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Probability Calculator

You can use this Probability Calculator to determine the probability of single and multiple events. Enter your values in the form and click the "Calculate" button to see the results.

Number of events occurred, n(E):

Number of possible outcomes, n(T):

Probability Formulas

The Single Event Probability Calculator uses the following formulas:

P(E) = n(E) / n(T) = (number of outcomes in the event) / (total number of possible outcomes)

P(E') = P(not E) = 1 - P(E)

P(E) is the probability that the event will occur,

P(E') is the probability that the event will not occur,

n(E) is the number of outcomes in the event E,

n(T) is the total number of possible outcomes.

Number of event occurs in A, n(A):

Number of event occurs in B, n(B):

The Multiple Event Probability Calculator uses the following formulas:

P(A) = n(A) / n(T)

P(A') = P(not A) = 1 - P(A)

P(B) = n(B) / n(T)

P(B') = P(not B) = 1 - P(B)

P(A ∩ B) = P(A) × P(B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A | B) = P(A ∩ B) / P(B)

P(B | A) = P(A ∩ B) / P(A)

P(A) is the probability that event A occurs,

P(A') is the probability that event A does not occur,

P(B) is the probability that event B occurs,

P(B') is the probability that event B does not occur,

P(A ∩ B) is the probability that events A and B both occur,

P(A ∪ B) is the probability that events A or B occur,

P(A | B) is the probability that event A occurs, given that event B has occurred,

n(A) is the number of outcomes in the event A,

n(B) is the number of outcomes in the event B,

You may also be interested in our Coin Flip Probability Calculator

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Probability Calculator

Probability of two events.

To find out the union, intersection, and other related probabilities of two independent events.

Probability Solver for Two Events

Please provide any 2 values below to calculate the rest probabilities of two independent events.

Probability of a Series of Independent Events

Probability of a normal distribution.

Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels.

Related Standard Deviation Calculator | Sample Size Calculator | Statistics Calculator

Probability is the measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 1 signifying certainty, and 0 signifying that the event cannot occur. It follows that the higher the probability of an event, the more certain it is that the event will occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the total number of outcomes. This is further affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both event A and B occur, and the probability that either event A or event B occurs, but not both.

Complement of A and B

Given a probability A , denoted by P(A) , it is simple to calculate the complement, or the probability that the event described by P(A) does not occur, P(A') . If, for example, P(A) = 0.65 represents the probability that Bob does not do his homework, his teacher Sally can predict the probability that Bob does his homework as follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would be calculated in the same manner, and it is worth noting that in the calculator above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35 , and can equal 0.30 or some other number.

Intersection of A and B

The intersection of events A and B , written as P(A ∩ B) or P(A AND B) is the joint probability of at least two events, shown below in a Venn diagram. In the case where A and B are mutually exclusive events, P(A ∩ B) = 0 . Consider the probability of rolling a 4 and 6 on a single roll of a die; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are independent. In this case, the probabilities of events A and B are multiplied. To find the probability that two separate rolls of a die result in 6 each time:

The calculator provided considers the case where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of event A given that event B has occurred, P(A|B) . Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B|A) = (3/10) × (7/9) = 0.2333

Union of A and B

In probability, the union of events, P(A U B) , essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that P(A U B) can also be written as P(A OR B) . In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event A is the probability that an even number is rolled, and event B is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so P(A U B) would be 3/6 + 3/6 = 1 , since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events A and B are not mutually exclusive. In this case:

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

Exclusive OR of A and B

Another possible scenario that the calculator above computes is P(A XOR B) , shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as P(A) = 0.65 , or Snickers being chosen with P(B) = 0.349 , and a P(unlikely) = 0.001 that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where μ is the mean and σ 2 is the variance. Note that standard deviation is typically denoted as σ . Also, in the special case where μ = 0 and σ = 1 , the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. P in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding P as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given μ = 68; σ = 4 (60 - 68)/4 = -8/4 = -2 (72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

Probability Calculator

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How does the conditional probability formula work? Let's say we had 2 events, A and B, and we wanted to calculate the probability of A given B, P(A|B). We could start by highlighting A, because we are looking at outcomes inside this circle. However, we have got more information to deal with in the question - we know that B happened. This means that we can exclude everything which is not in B, since we know that we are looking at outcomes where B happened. We can represent this in a Venn diagram as follows:

Bayes' Theorem

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Probability Calculator

The probability calculator can find two events' probability and the normal distribution probability. Learn more about probability's laws and calculations.

Probability of A: P(A)

Probability of B: P(B)

Probability of

Event A Probability

Repeat Times

Event B Probability

Standard Deviation (σ)

Left Bound (Lb)

Right Bound (Rb)

Related Calculators

Probability

Probability of A: P(A) = 0.5 Probability of B: P(B) = 0.4 Probability of A NOT occuring: P(A') = 1 - P(A) = 0.5 Probability of B NOT occuring: P(B') = 1 - P(B) = 0.6 Probability of A and B both occuring: P(A∩B) = P(A) × P(B) = 0.2 Probability that A or B or both occur: P(A∪B) = P(A) + P(B) - P(A∩B) = 0.7 Probability that A or B occurs but NOT both: P(AΔB) = P(A) + P(B) - 2P(A∩B) = 0.5 Probability of neither A nor B occuring: P((A∪B)') = 1 - P(A∪B) = 0.3 Probability of A occuring but NOT B: P(A) × (1 - P(B)) = 0.3 Probability of B occuring but NOT A: (1 - P(A)) × P(B) = 0.2

Probability of A occuring 5 time(s) = 0.6 5 = 0.07776 Probability of A NOT occuring = (1-0.6) 5 = 0.01024 Probability of A occuring = 1-(1-0.6) 5 = 0.98976 Probability of B occuring 3 time(s) = 0.3 3 = 0.027 Probability of B NOT occuring = (1-0.3) 3 = 0.343 Probability of B occuring = 1-(1-0.3) 3 = 0.657 Probability of A occuring 5 time(s) and B occuring 3 time(s) = 0.6 5 × 0.3 3 = 0.00209952 Probability of neither A nor B occuring = (1-0.6) 5 × (1-0.3) 3 = 0.00351232 Probability of both A and B occuring = (1-(1-0.6) 5 ) × (1-(1-0.3) 3 ) = 0.65027232 Probability of A occuring 5 times but not B = 0.6 5 × (1-0.3) 3 = 0.02667168 Probability of B occuring 3 times but not A = (1-0.6) 5 × 0.3 3 = 2.7648e-4 Probability of A occuring but not B = (1-(1-0.6) 5 ) × (1-0.3) 3 = 0.33948768 Probability of B occuring but not A = (1-0.6) 5 × (1-(1-0.3) 3 ) = 0.00672768

The probability between -1 and 1 is 0.68268 The probability outside of -1 and 1 is 0.31732 The probability of -1 or less (≤-1) is 0.15866 The probability of 1 or more (≥1) is 0.15866

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Table of Contents

Probability of two events calculator, probability solver for two events, probability of a series of independent events, probability of a normal distribution, introduction to probability, rules of event operations, complement an event, intersection of events, independent events, union of events, normal distribution, probability of normal distribution.

When you know the probability of two independent events, you can use the Probability of Two Events Calculator to determine their occurring together. You have to enter the probabilities of two independent events as the probability of a and b in the calculator. Then the calculator will show the union, intersection, and other related probabilities of two independent events along with the Venn diagrams.

You can compute the probability of various events of two independent events if you know any two input values of the Probability Solver for Two Events Calculator. This is important when you do not have one or both probabilities of two events. The results will show the answer with the calculation steps.

You can use the Probability of a Series of Independent Events Calculator to determine the probability of when each experiment contains two independent events that happen one after the other. In this calculator, you must set the number of times the event occurs.

The normal distribution probability calculator is helpful when determining the probability of a normal curve. You must insert the mean μ , standard deviation σ , and boundaries. The normal probability calculator will generate the probability of the set boundaries and the confidence intervals for a range of confidence levels.

Probability is the chance that an event will happen. When an event is unquestionably going to happen, its probability is 1. When an event is not going to happen, its probability is 0. As a result, a given event's probability is always between 0 and 1. The probability calculator makes calculating probabilities for various events incredibly simple.

Any grouping of an experiment's results is referred to as an event. It is an event that can be any subset of the sample space. The complement, intersection, and union can be identified as rules of event operations. Let's learn each of these rules using the below example.

Your college has various faculties, including business faculty. International students are also enrolled in this college. You must conduct interviews with your college students as part of your project. You choose to begin with the first student who walks through the gate. You are aware of the following probabilities. Let's say,

A = The first student is from the Business Faculty.

B = The first student is an international student.

The complement of an event is the set of all outcomes in a sample space that are not included in that event.

For example, the complement of event A means the first student is from somewhere other than the business faculty. This can be denoted by \$A\prime\$ or Aᶜ.

Let's show the complement of event A in a Venn diagram.

In the above Venn diagram, the colored area represents the complement of event A.

The rectangle's total area represents the overall probability of the sample space. It is precisely one. The space outside circle A shows the probability of the complement of event A. The Venn diagram allows us to establish the following relationship:

$$P\left(A\right)+P\left(A^\prime\right)=1$$

$$P\left(A^\prime\right)=1-P\left(A\right)$$

Let's find the following probabilities.

The probability of the first student you are selecting for the interview is not from the business faculty:

$$P\left(A^\prime\right)=1-P\left(A\right)=1-0.6=0.4$$

The probability of the first student you select for the interview is not an international student:

$$P\left(B^\prime\right)=1-P\left(B\right)=1-0.3=0.7$$

The intersection of two events A and B is the list of all common elements in both events A and B. The word "AND" is frequently used to indicate the intersection of two sets.

The intersection of event A and event B in example 1 means selecting an international student, and the student is from the business faculty. This can be denoted as follows:

$$A\cap B$$

Let's show the intersection of events A and B in a Venn diagram.

In the above Venn diagram, the colored area represents the intersection of events A and B.

Let's say the event of selecting a local student for the interview is C. Now, we will show events A and C in a Venn diagram.

Selecting an international student and a local student cannot be done simultaneously. Suppose the first student you choose is an international student. In that case, it excludes the event of the first student being a local student. Therefore, events A and C are mutually exclusive events.

The mutually exclusive events do not have any common elements between them. Therefore, the intersection of two mutually exclusive events is empty.

$$A\cap C=φ$$

The probability of events' intersection can be calculated with different methods. Events A and B can be written as follows.

$$P\left(A\cap B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cup B\right)$$

$$P\left(A\cap B\right)=P(A)× P(B/A)$$

$$P\left(A\cap B\right)=P(B)× P(A/B)$$

Independent events are events that do not influence one another. In our example, selecting a student from the business faculty does not affect choosing an international student or not. Therefore, we can say that event A and event B are two independent events.

When events are independent, the probability of any one of them happening does not depend upon that of the other. Therefore,

$$P(B/A)=B\ and\ P(A/B)=A$$

You can use these formulas to modify the formula we previously learned to determine the probability of two intersection events.

$$P\left(A\cap B\right)=P\left(A\right)× P\left(\mathrm{B/A}\right)P\left(A\cap B\right)=P(A)× P(B)$$

$$P\left(A\cap B\right)=P\left(B\right)× P\left(\mathrm{A/B}\right)P\left(A\cap B\right)=P(B)× P(A)$$

Therefore, you can find the intersection of the two independents by multiplying the probability of those two events.

$$P\left(A\cap B\right)=P\left(A\right)× P\left(B\right)=P(B)× P(A)$$

Given that events A and B are independent let's determine the probability that the first student you select for the interview will be from the business faculty and be an international student.

$$P\left(A\cap B\right)=P\left(A\right)× P\left(B\right)=0.6× 0.3=0.18$$

The union of two events produces another event that contains all elements from either or both events. The word "OR" is typically used to describe the union of two events.

In Example 1, the union of events A and B means selecting an international student or a student from the business faculty. This can be denoted as follows.

$$A\cup B$$

Let's show the union of events A and B in a Venn diagram.

The above Venn diagram colored area represents the union of events A and B.

To calculate the probability of event A or event B, we must add the probabilities of both events and subtract the probability of the intersection.

The probability of a union of events A and B can be written as follows.

$$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$$

We can modify the above formula and create a new formula to find the probability of the union of two independent events when the probability of the intersection of two events is unknown and the two events are independent.

If the events are independent,

$$P\left(A\cap B\right)=P(A)× P(B)$$

$$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P(A)× P(B)$$

Let's calculate what would be the probability of combining events A and B, that is, with what probability would we choose a student who is a business major, an international student, or both at the same time?

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.6+0.3-0.18=0.72$$

Thanks to the Probability of Two Events Calculator or Probability Solver for Two Events Calculator, you can complete all the calculations above quickly. You can use the Probability Solver for Two Events Calculator even if you would like to check your probability calculation steps because it also displays the steps for the calculation.

The normal distribution is symmetrical and has a bell shape. A normal distribution has an identical mean, median, and mode as well as 50% of the data above the mean and 50% below the mean. The normal distribution curve goes away from the mean in both directions but never touches the X-axis. The total area under the curve is 1.

If random variable X has a normal distribution with parameters μ and σ2, we write X ~ N(μ, σ²) .

A normal distribution's probability density function is depicted below:

$$f\left(x\right)=\frac{1}{\sqrt{2π\sigma^2}}× e^\frac{-{(x-\mu)}^2}{2\sigma^2}$$

In this function:

• μ is the mean of the distribution;
• σ² is the variance of the distribution;
• e is 2.7182.

It is impossible to provide a probability table for each combination of mean and standard deviation because there are an infinite number of different normal curves. The standard normal distribution is utilized as a result. The normal distribution with a mean of 0 and a standard deviation of 1 is referred to as the standard normal distribution.

To calculate the probability of a normal distribution, we must first transform the actual distribution into a standard normal distribution using the z-score and then use the z-table to calculate the probability. The normal probability calculator functions as a standard normal probability calculator by offering probabilities for various confidence levels.

$$Z=\frac{X-\mu}{\sigma}$$

The standard normal distribution curve can be used to resolve a variety of real world problems. To determine the probability of continuous variables, the normal distribution is utilized. A continuous variable is a variable that can assume any number of values, even a decimal. A few examples of continuous variables are height, weight, and temperature.

Let's learn how to find the probability of normal distribution using the below example.

Your batch's statistics course results are normally distributed, with a mean of 65 and a standard deviation of 10. Determine the probability of the following scenarios if a student is selected at random:

• the student's score is equal or above 70,
• the student's score is less than 70,
• the student's score is between 50 and 70.

$$P\left(X≥70\right)=P\left(Z≥\frac{70-65}{10}\right)=P\left(Z≥0.5\right)=1-0.6915=0.3085$$

$$P\left(X<70\right)=P\left(Z<\frac{70-65}{10}\right)=P\left(Z<0.5\right)=0.6915$$

$$P\left(50>X>70\right)=P\left(\frac{50-65}{10}>Z>\frac{70-65}{10}\right)=P\left(1.5>Z>0.5\right)=0.4332+0.1915=0.6247$$

Computing the probability of a normal curve involves numerous steps and requires using z-tables. On the other hand, the normal distribution probability calculator helps you compute probability simply by entering four numbers into the calculator. To use the normal distribution calculator, you only need to enter the mean, standard deviation, and left and right boundaries.

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Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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Probability

Probability, at its core, is an area of mathematics where uncertainty and randomness are carefully studied. This field revolves around the art of measuring the chances of events or outcomes occurring in different scenarios. Within probability theory, events are typically depicted as subsets of a sample space, with probabilities assigned to these events to convey their expected occurrence.

Key concepts in probability include:

• Probability Space: A probability space consists of a sample space (all possible outcomes) and a probability measure (assigning probabilities to events in the sample space).
• Random Variables: Random variables are mathematical representations of uncertain quantities. They can take different values with corresponding probabilities.
• Probability Distributions: Probability distributions describe the probabilities of different outcomes for a random variable. Common distributions include the normal distribution, binomial distribution, and Poisson distribution.
• Conditional Probability: Conditional probability calculates the probability of an event occurring given that another event has already occurred.
• Independence: Events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of another event occurring or not occurring.

Statistics involves collecting, studying, interpreting, organizing, and presenting data, all with the goal of obtaining significant information and making informed judgments based on observed data. This field serves as a tool for summarizing information, identifying underlying patterns, and extrapolating knowledge about broader populations or phenomena through the analysis of representative sample data.

Key concepts in statistics include:

• Descriptive Statistics: Descriptive statistics include methods for summarizing and presenting data. Common descriptive measures include mean (average), median, mode, range, and standard deviation.
• Inferential Statistics: Inferential statistics use sample data to make inferences about the larger population. Methods such as hypothesis testing, confidence intervals, and regression analysis fall into this category.
• Sampling: Sampling is the process of selecting a subset (sample) from a larger group (population) in order to draw conclusions about the entire population.
• Statistical Distributions: Understanding the properties of statistical distributions is critical to statistical inference. Common distributions include the normal distribution and the t-distribution.
• Statistical Tests: Statistical tests are used to evaluate whether observed differences or relationships between variables are statistically significant or due to chance.

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This calculator finds the nth root of the product of $$$ n $$$ numbers.

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Hypergeometric Distribution

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This calculator quickly identifies the central value in a data set, giving a true midpoint even when outliers are present.

Useful in market research and social sciences, it pinpoints the value or values that appear most frequently in a data set.

Normal Distribution

It calculates probabilities related to the bell curve, essential for various applications.

In hypothesis testing, this calculator evaluates the statistical significance, determining the strength of evidence against a null hypothesis.

Pearson Correlation Coefficient

Analyzing linear relationships between two variables, this tool measures their strength and direction, aiding in predictive analytics.

This is a tool that offers a relative ranking, showing where a particular score stands compared to others in a data set.

Percentile and Class Rank

Designed mainly for educational metrics, it ranks a student's score relative to their peers or a specific distribution.

Sample and Population Coefficients of a Variation

This tool measures the relative variability of data, offering insight into the stability of the data regardless of the mean.

Sample and Population Covariances

Determining how two sets of data change together is critical to understanding the relationships between variables.

Sample and Population Standard Deviation

A fundamental tool in statistics, it quantifies the amount by which individual data points deviate from the mean.

Sample and Population Variance

By squaring standard deviations, this calculator provides insight into data dispersion in a set or population.

Upper Quartile

Giving a perspective on the upper half of the data, it determines the 75th percentile or the median of the upper half.

Indispensable in standardizing scores, this tool measures how many standard deviations a value is from the mean.

5-Number Summary

Providing a complete overview of the data, it gives the minimum, lower quartile, median, upper quartile, and maximum, making it easy to interpret the data.

What does the Probability and Statistics Calculator offer?

It offers a wide collection of calculators to perform different statistical and probabilistic computations. Whether you're dealing with elementary statistics or advanced data analysis, this calculator has got you covered. Whether you are dealing with elementary statistics or advanced data analysis, this calculator will help you.

How is the Statistics Calculator different from the probability calculator?

The Statistics Calculator focuses primarily on data analysis and descriptive statistics. In contrast, a Probability Calculator specializes in predicting the likelihood of specific outcomes, such as the likelihood of three certain events occurring simultaneously.

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Probability - Problem Solving

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To solve problems on this page, you should be familiar with

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Problem Solving - Basic

Problem solving - intermediate, problem solving - difficult.

If I throw 2 standard 5-sided dice, what is the probability that the sum of their top faces equals to 10? Assume both throws are independent to each other. Solution : The only way to obtain a sum of 10 from two 5-sided dice is that both die shows 5 face up. Therefore, the probability is simply \( \frac15 \times \frac15 = \frac1{25} = .04\)

If from each of the three boxes containing \(3\) white and \(1\) black, \(2\) white and \(2\) black, \(1\) white and \(3\) black balls, one ball is drawn at random. Then the probability that \(2\) white and \(1\) black balls will be drawn is?

2 fair 6-sided dice are rolled. What is the probability that the sum of these dice is \(10\)? Solution : The event for which I obtain a sum of 10 is \(\{(4,6),(6,4),(5,5) \}\). And there is a total of \(6^2 = 36\) possible outcomes. Thus the probability is simply \( \frac3{36} = \frac1{12} \approx 0.0833\)

If a fair 6-sided dice is rolled 3 times, what is the probability that we will get at least 1 even number and at least 1 odd number?

Three fair cubical dice are thrown. If the probability that the product of the scores on the three dice is \(90\) is \(\dfrac{a}{b}\), where \(a,b\) are positive coprime integers, then find the value of \((b-a)\).

You can try my other Probability problems by clicking here

Suppose a jar contains 15 red marbles, 20 blue marbles, 5 green marbles, and 16 yellow marbles. If you randomly select one marble from the jar, what is the probability that you will have a red or green marble? First, we can solve this by thinking in terms of outcomes. You could draw a red, blue, green, or yellow marble. The probability that you will draw a green or a red marble is \(\frac{5 + 15}{5+15+16+20}\). We can also solve this problem by thinking in terms of probability by complement. We know that the marble we draw must be blue, red, green, or yellow. In other words, there is a probability of 1 that we will draw a blue, red, green, or yellow marble. We want to know the probability that we will draw a green or red marble. The probability that the marble is blue or yellow is \(\frac{16 + 20}{5+15+16+20}\). , Using the following formula \(P(\text{red or green}) = 1 - P(\text{blue or yellow})\), we can determine that \(P(\text{red or green}) = 1 - \frac{16 + 20}{5+15+16+20} = \frac{5 + 15}{5+15+16+20}\).

Two players, Nihar and I, are playing a game in which we alternate tossing a fair coin and the first player to get a head wins. Given that I toss first, the probability that Nihar wins the game is \(\dfrac{\alpha}{\beta}\), where \(\alpha\) and \(\beta\) are coprime positive integers.

Find \(\alpha + \beta\).

If I throw 3 fair 5-sided dice, what is the probability that the sum of their top faces equals 10? Solution : We want to find the total integer solution for which \(a +b+c=10 \) with integers \(1\leq a,b,c \leq5 \). Without loss of generality, let \(a\leq b \leq c\). We list out the integer solutions: \[ (1,4,5),(2,3,5), (2,4,4), (3,3,4) \] When relaxing the constraint of \(a\leq b \leq c\), we have a total of \(3! + 3! + \frac{3!}{2!} + \frac{3!}{2!} = 18 \) solutions. Because there's a total of \(5^3 = 125\) possible combinations, the probability is \( \frac{18}{125} = 14.4\%. \ \square\)

Suppose you and 5 of your friends each brought a hat to a party. The hats are then put into a large box for a random-hat-draw. What is the probability that nobody selects his or her own hat?

How many ways are there to choose exactly two pets from a store with 8 dogs and 12 cats? Since we haven't specified what kind of pets we pick, we can choose any animal for our first pick, which gives us \( 8+12=20\) options. For our second choice, we have 19 animals left to choose from. Thus, by the rule of product, there are \( 20 \times 19 = 380 \) possible ways to choose exactly two pets. However, we have counted every pet combination twice. For example, (A,B) and (B,A) are counted as two different choices even when we have selected the same two pets. Therefore, the correct number of possible ways are \( {380 \over 2} = 190 \)

A bag contains blue and green marbles. If 5 green marbles are removed from the bag, the probability of drawing a green marble from the remaining marbles would be 75/83 . If instead 7 blue marbles are added to the bag, the probability of drawing a blue marble would be 3/19 . What was the number of blue marbles in the bag before any changes were made?

Bob wants to keep a good-streak on Brilliant, so he logs in each day to Brilliant in the month of June. But he doesn't have much time, so he selects the first problem he sees, answers it randomly and logs out, despite whether it is correct or incorrect.

Assume that Bob answers all problems with \(\frac{7}{13}\) probability of being correct. He gets only 10 problems correct, surprisingly in a row, out of the 30 he solves. If the probability that happens is \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, find the last \(3\) digits of \(p+q\).

Out of 10001 tickets numbered consecutively, 3 are drawn at random .

Find the chance that the numbers on them are in Arithmetic Progression .

The answer is of the form \( \frac{l}{k} \) .

Find \( k - l \) where \(k\) and \(l\) are co-prime integers.

HINT : You might consider solving for \(2n + 1\) tickets .

You can try more of my Questions here .

A bag contains a blue ball, some red balls, and some green balls. You reach into​ the bag and pull out three balls at random. The probability you pull out one of each color is exactly 3%. How many balls were initially in the bag?

More probability questions

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Amanda decides to practice shooting hoops from the free throw line. She decides to take 100 shots before dinner.

Her first shot has a 50% chance of going in.

But for Amanda, every time she makes a shot, it builds her confidence, so the probability of making the next shot goes up, But every time she misses, she gets discouraged so the probability of her making her next shot goes down.

In fact, after \(n\) shots, the probability of her making her next shot is given by \(P = \dfrac{b+1}{n+2}\), where \(b\) is the number of shots she has made so far (as opposed to ones she has missed).

So, after she has completed 100 shots, if the probability she has made exactly 83 of them is \(\dfrac ab\), where \(a\) and \(b\) are coprime positive integers, what is \(a+b\)?

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Tool to calculate the birthday paradox problem in probabilities. How many people are necessary to have a 50% chance that 2 of them share the same birthday.

Birthday Probabilities - dCode

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Probability to share a same birthday

Answers to questions (faq).

• What is the birthday paradox? (Definition)

The birthday paradox is a mathematical problem put forward by Von Mises. It answers the question: what is the minimum number $ N $ of people in a group so that there is a 50% chance that at least 2 people share the same birthday (day-month couple). The answer is $ N = 23 $, which is quite counter-intuitive, most people estimate this number to be much larger, hence the paradox.

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not true in reality, but it's close).

In the following FAQ, a year has 365 days (leap years are ignored).

• What is the probability for a person to be born on a given day of the year?

Probability is $ 1/365 \approx 0.0027 \approx 0.27\% $, indeed, 1 chance out of 365 to be born on a precise day in a year with 365 days.

Example: A random person has a 0.27% chance of being born on April 1st (or any other day of the year).

Example: Any average human has a 0.27% chance of being born on the same day as me/you.

NB: the complementary (opposite) probability of not being born on a certain day of the year is $ 1-1/365 = 364/365 \approx 0.9973 \approx 99.73\% $, indeed, there are 364 possible days out of 365.

Example: A random person has a 99.73% chance of not being born on August 15 (or any other day of the year).

Example: Any average human has a 99.73% chance of not being born on the same day as me/you.

• What are the odds for 2 people to be born the same day?

By taking 2 people at random, and noting them A and B, this calculation amounts to asking the question What is the probability that B was born on a certain day of the year? this certain day being the birthday of birth of A. The probability is always $ 1/365 \approx 0.0027 \approx 0.27\% $

Example: The probability that a mother was born on the same day as her child is 0.27%

Example: The probability that a parent has a second child born on the same day (not the same year) as the first child is 0.27%

To calculate the odds for 2 people to be born on different days, take the opposite, so $ 1 - 1/365 = 364/365 \approx 0.9973 \approx 99.73\% $

• What are the odds for 2 people in a group of N to be born the same day?

Calculating this probability is equivalent to calculating the opposite of the probability that all people were born on a different day (because in this case at least 2 would be born on the same day).

For the smallest of the groups: N = 2. This calculation is explained above in What is the probability that 2 people were born on the same day? so $ P(N=2) \approx 0.27\% $. Note that this probability is the opposite of the probability that a person A was born on a different day from a person B and can therefore also be calculated $$ P(N=2) = 1 - (364/365) = 0.0027 = 0.27\% $$

For a larger group, N=3 composed of people A, B and C. There are therefore 364 chances out of 365 that B was not born on the same day as A and 363 chances out of 365 that C was not born the same days as A and B. Thus, $$ P(N=3) = 1 - (364/365) \times (363/365) \approx 0.82\% $$

In the same way, $$ P(N=4) = 1 - (364/365) \times (363/365) \times (362/365) \approx 1.64\% \\ P(N=5) = 1 - (364/365) \times \cdots \times (361/365) \approx 2.71\% $$

The general formula is $$ P(N=n) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

A group of 23 people has a probability of approximately 0.5 or 50% (1 chance out of 2) that two people have a common birth date (day + month).

$$ P(N=23) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{343}{365}\right) \approx 0.5073 $$

• What are the odds for no one in a group of N to be born the same day?

The probability that no one shares a birthday is the opposite of the probability that there are (at least) 2 in common

$$ P(N=n) = \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

• What is the probability for N people to be born the same day?

By taking N people at random numbered from 1 to N, then by denoting D the date of birth of the first person, this amounts to calculating the probability that N-1 other people were born on date D. For person 2, the probability is $ P= 1/365 \approx 0.0027 \approx 0.27\% $, for person 3, same proba, $ P=1/365 $, etc. The probabilities are multiplied , i.e. the formula:

$$ P(X=N) = \left( \frac{1}{365} \right)^{N-1} $$

Example: For $ N = 3 $ people, $ P(X=3) = 1/(365^2) \approx 0.0000075 \approx 0.00075\% $ or 1 chance out of (365*365)=133225

Example: The probability that a mother and her 4 children (so 5 people) are born on the same day is 1 in 365^4

• What is the probability of at least 2 people among N to be born on a given day?

Calculate the opposite of the probability that the N people were born on another day.

$$ P(N=n) = 1 - (364/365)^(n-1) $$

Example: $ P(N=3) = 1 - (364/365)^2 \approx 0,0055 \approx 0.55\% $

For a probability of 50%, a minimum of 253 people will be needed that the probability that 2 were born on a specific day is approximately 1/2.

Example: $ P(N=253) = 1 - (364/365)^252 \approx 0.499105 \approx 50\% $

NB: The complementary probability that all N people are not born on a given date is P = (364/365)^N

• How many people are needed in a group to be sure that 2 share the same birthday?

366 are needed, ie, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

The answer would be 367 taking leap years into account.

• How many people are born the same day as myself?

With the hypothesis of a world population of 8 billion people.

People born on February 29 represent statistically 0.06653% of the population or 5 million people and are therefore negligible.

There are 8000000000/365 or about 22 million people born on a given day+month.

Limiting to the USA (350 millions inhabitants), there are 350000000/365 or nearly 1 millions US people born on a given day+month date.

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Unit 7: Probability

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Probability tells us how often some event will happen after many repeated trials. You've experienced probability when you've flipped a coin, rolled some dice, or looked at a weather forecast. Go deeper with your understanding of probability as you learn about theoretical, experimental, and compound probability, and investigate permutations, combinations, and more!

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Multiplication rule for independent events

• Sample spaces for compound events (Opens a modal)
• Compound probability of independent events (Opens a modal)
• Probability of a compound event (Opens a modal)
• "At least one" probability with coin flipping (Opens a modal)
• Free-throw probability (Opens a modal)
• Three-pointer vs free-throw probability (Opens a modal)
• Probability without equally likely events (Opens a modal)
• Independent events example: test taking (Opens a modal)
• Die rolling probability with independent events (Opens a modal)
• Probabilities involving "at least one" success (Opens a modal)
• Sample spaces for compound events Get 3 of 4 questions to level up!
• Independent probability Get 3 of 4 questions to level up!
• Probabilities of compound events Get 3 of 4 questions to level up!
• Probability of "at least one" success Get 3 of 4 questions to level up!

Multiplication rule for dependent events

• Dependent probability introduction (Opens a modal)
• Dependent probability: coins (Opens a modal)
• Dependent probability example (Opens a modal)
• Independent & dependent probability (Opens a modal)
• The general multiplication rule (Opens a modal)
• Dependent probability (Opens a modal)
• Dependent probability Get 3 of 4 questions to level up!

Conditional probability and independence

• Calculating conditional probability (Opens a modal)
• Conditional probability explained visually (Opens a modal)
• Conditional probability using two-way tables (Opens a modal)
• Conditional probability tree diagram example (Opens a modal)
• Tree diagrams and conditional probability (Opens a modal)
• Conditional probability and independence (Opens a modal)
• Analyzing event probability for independence (Opens a modal)
• Calculate conditional probability Get 3 of 4 questions to level up!
• Dependent and independent events Get 3 of 4 questions to level up!

Statistics Calculators

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Top statistics calculators

The critical value calculator helps you find the one- and two-tailed critical values for the most widespread statistical tests.

The p-value calculator can help you find the p-value and evaluate how compatible your data is with the null hypothesis.

The confidence interval calculator finds the confidence level for your data sample.

Use the probability calculator to find the likelihood of various interactions between two distinct events.

t-test calculator performs all kinds of t-tests: one-sample, two-sample, and paired.

With this dice probability calculator, you can easily find the various probabilities related to rolling a set of dice.

The z-score calculator can help you determine the standard score for a data point.

The empirical rule calculator allows you to find the three intervals within which you'll find 68, 95, and 99.7% of your data.

Use the combinations calculator to determine the number of combinations for a set.

Coin flip probability calculator lets you calculate the likelihood of obtaining a set number of heads when flipping a coin multiple times.

Use the odds calculator to convert odds to a probability of winning or losing.

The degrees of freedom calculator finds the number of independent values in a data sample.

Use the binomial distribution calculator to calculate the probability of a certain number of successes in a sequence of experiments.

Our Z-test calculator is here to help you learn about, and perform, a one-sample Z-test.

Point estimate calculator computes the "best guess" of an unknown population parameter.

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The accuracy calculator is a simple tool for calculating accuracy using three simple methods.

Bayes' theorem calculator helps calculate conditional probabilities in accordance with Bayes' rule.

What is the chance of choosing the box containing only gold? Find the answer to Bertrand's box paradox with Omni!

Bertrand's paradox is an intriguing warning for every scientist: dealing with infinity and randomness can lead to pitfalls!

The birthday paradox calculator is a tool that enables you to determine the probability that at least two people from a group of a given size will share a birthday.

The boy or girl paradox is an excellent example of why how we ask questions matters!

The Chebyshev's theorem calculator counts the probability of an event being far from its expected value.

With this coin toss streak calculator, you will discover a very interesting problem in probability related to consecutive heads appearing in coin flips.

Use the conditional probability calculator to determine the probability of an event conditional on another event.

Our confusion matrix calculator helps you to calculate all the metrics you need to assess the performance of your machine learning model.

With this dice average calculator, you can quickly calculate the average value of a set of dice rolls.

Use this dice roller calculator when you've lost the dice to your favorite board game. Throw up to 15 dice of twenty different types.

Unlock the power of statistics with our expected value formula calculator. Learn how to calculate the expected value swiftly. Try it today!

Discover why "accurate" tests can be misleading with this false positive paradox calculator.

Our implied probability calculator helps you to calculate the probability of an incident happening, given the odds.

The lottery calculator finds the odds for winning in a typical lottery.

The Monty Hall Problem calculator can help you win a car 66% of the time — but only if you play the game!

Learn how two losing games can make a winning game by alternating between them in this Parrondo's paradox calculator.

Use Omni's password combinations calculator to find how many possible combinations of numbers, letters, and symbols exist, given the length of a password.

Use the permutation calculator to determine the number of permutations in a set.

Use this p-hat calculator to determine the sample proportion according to the number of occurrences of an event and the sample size.

The post-test probability calculator allows you to calculate pre-test probability, likelihood ratios, and finally, the post-test probability itself.

The probability of 3 events calculator determines the chances of three independent events occurring.

No more straining with the tossing a coin and the rolling of dice - this random number generator will give you the figures you need.

Use the relative risk calculator to compare the probability of developing a disease in two groups of people.

Risk calculator checks which of the two options is less risky based on the probability of failure and its consequences.

Find your bet's odds and potential winnings with the roulette payout calculator!

Our sensitivity and specificity calculator is a simple way to calculate all the values needed in medical statistics.

The two envelopes paradox calculator helps you determine whether swapping envelopes will make you obtain the envelope with a larger amount of money.

Benford's law calculator allows you to verify if your data set follows Benford's distribution of leading digits.

Beta distribution calculator can evaluate all things related to this famous family of probability distributions.

The box plot calculator is here to show you a graphic analysis of your dataset in the form of a box-and-whisker plot.

The central limit theorem calculator allows you to calculate the sample mean and the sample standard deviation for the given population distribution and sample size.

Determine the goodness of fit of your data with the chi-square calculator.

Calculate the class width of any dataset.

The continuity correction calculator shows you how to apply continuity correction to approximate a binomial distribution of a given problem.

Make your own dot plots with this dot plot calculator!

The exponential distribution calculator finds out the probability of a certain amount of time elapsing between two events.

Exponential growth prediction tool estimates a future value based on past performances.

The frequency distribution calculator generates the cumulative frequency distribution table and a bar graph representing the frequency distribution for the given set of numbers.

The frequency polygon calculator will help you create a frequency polygon for the given set of data points.

Our geometric distribution calculator will help you determine the probability of a certain number of trials needed for success.

The histogram calculator is a histogram maker and a lesson on histograms, all in one. Let's explore what is a histogram, some examples, and the differences between a histogram vs a bar chart.

Use the hypergeometric distribution calculator to find the probability (or cumulative probability) associated with the hypergeometric distribution.

Our lognormal distribution calculator helps you solve problems regarding probabilities, quantiles, samples, etc. from the lognormal distribution.

The negative binomial distribution calculator calculates what is the probability according to the Pascal distribution.

Effortlessly calculate probabilities with our bell curve calculator, a perfect tool for any probability distribution analysis. Try it today!

The normal probability calculator for sampling distributions gives you the probability of finding a range of sample mean values.

Using this pie chart calculator, you can compare values in different groups visually using a pie chart.

The Poisson distribution calculator is a tool for determining the probability of a certain number of independent events happening in a given time.

Our Rayleigh distribution calculator helps you generate samples from the Rayleigh distribution, or determine probabilities, cdf, pdf, quantiles, and common measures in this distribution.

Our relative frequency calculator can help you find the experimental probability for ungrouped and grouped data, as well as cumulative relative frequency.

Use our sampling distribution of the sample proportion calculator to find the probability that your sample proportion falls within a range.

SMp(x) distribution calculator can generate a six-parameter probability function, which can simulate virtually every known probability distribution.

Convert any given set of integers to a stemplot using this stem-and-leaf plot calculator!

The uniform distribution calculator can perform all sorts of computations related to this popular probability distribution. Comes with free sample generator!

The upper control limit calculator helps you determine the lower and upper control limit values of your dataset.

Use this Weibull distribution calculator if you need find probabilities, quantiles, etc. in the Weibull distribution, or generate samples from this distribution.

The 5 number summary calculator will give you the five most important statistical parameters of your dataset: minimum, first quartile, median, third quartile, and maximum.

The rating average calculator returns an average rating based on the number of 1-5 ratings.

The coefficient of variation calculator is a convenient way to describe the standard deviation as a percentage of the mean.

Our Cohen's D calculator can help you measure the standardized effect size between two data sets.

Our constant of proportionality calculator can help you to calculate the ratio that relates two dependable given values.

Correlation coefficient calculator quickly finds the values of several different correlation coefficients. If you wish, it will show you the intermediate steps of the calculations as well!

The correlation coefficient calculator helps you determine the statistical significance of your data with the Matthews correlation formula.

Covariance calculator gives you the sample covariance for two equally sized samples, as well as an estimate of population covariance.

Using the decile calculator, you can divide your data into deciles in order to interpret them.

The descriptive statistics calculator will thoroughly analyze your data set using more than 20 different descriptive statistics spread over various subcategories of statistical measurements, such as central tendency and dispersion.

The dispersion calculator is a handy tool that calculates the spread of data using multiple measures like range, interquartile range, variance, and standard deviation.

This error propagation calculator helps propagate the primary parameter(s)' error to the final parameter.

Use the grouped data standard deviation calculator to find the mean, variance, and standard deviation of given data ranges.

Calculate the index of qualitative variation easily with our tool, employing the IQV formula for accurate and fast analysis.

The IQR calculator allows you to find the interquartile range of up to 50 values.

The least to greatest calculator is here to put your numbers (up to fifty of them) in ascending order, even if instead of specific values, you give it arithmetic expressions.

The mean absolute deviation calculator can calculate the mean absolute deviation around the mean, median, or any other statistically significant number.

The mean median mode calculator tells you the mean, median, mode, range, and midrange of a dataset. We also show you all the steps!

The median calculator allows you to calculate the median number of a dataset with up to 50 values.

The median absolute deviation calculator can calculate the median absolute deviation of a data set of up to 50 points.

The midrange calculator will quickly calculate the midrange of a given set of data.

The minimum and maximum calculator will quickly give you the smallest and largest values in your dataset of up to fifty entries.

The mode calculator finds the modal value for your dataset.

The MSE calculator can quickly find the mean squared error of your sample and provide you with intermediate calculations. SSE and RMSE are available as well!

The outlier calculator is here to analyze your dataset of up to thirty entries and tell you if any of them are outliers, i.e., differ a lot from the others.

Use this Pearson correlation calculator to find Pearson's r of any given dataset, as well as a general oversight on what Pearson's correlation is all about.

The percentile calculator is here to help you get the k-th percentile of a data set of up to 30 numbers.

This percentile rank calculator finds the percentile for any data value in a dataset that contains up to 30 numbers.

Our pooled standard deviation calculator can help you measure the variability or spread of data when combining or pooling multiple datasets.

Use the population variance calculator to estimate the variance of a given population from its sample.

Our process capability index calculator helps you calculate whether the variation of your process is within specification limits and whether your process can produce the intended output.

The quartile calculator is a simple tool to help you find the quartiles of your dataset (and some additional extra information).

The range calculator will quickly calculate the range of a given data set.

The relative standard deviation calculator presents the standard deviation as a percentage of the mean.

Efficiently calculate diversity index and understand the Gini index formula with our Simpson's diversity index calculator. Try it today!

This skewness calculator finds both the skewness and kurtosis of a dataset and interprets these values, telling you how skewed or peaked your distribution is.

Spearman's correlation calculator can help you determine the value of this popular measure of rank association between two variables.

The standard deviation calculator tells you the mean and standard deviation of a dataset.

Here you can find the standard deviation index calculator and its possible uses for your test model.

Use this tool to calculate the standard deviation of the sample mean, given the population standard deviation and the sample size.

Use this standard error calculator whenever you need to find the standard error of the mean of any dataset with up to 30 numbers.

Use this sum of squares calculator to determine the variability of your data.

The upper and lower fence calculator determines the cutoff points for outliers in a dataset with up to 50 values.

The variance calculator finds the variance of a set of numbers. It is a measure of how tightly clustered the data is around the mean.

Discover the ultimate Venn diagram solver with our intuitive calculator. Effortlessly visualize complex data intersections and unions. Try it today!

Find how far the measured value may be from the real one using the absolute uncertainty calculator.

Determine the statistical significance of your A/B tests effortlessly with our AB test calculator. Reliable, quick results. Start calculating now!

This coefficient of determination calculator finds the R-squared of any bivariate dataset, and provides an interpretation as well!

This cubic regression calculator will help you determine the polynomial of degree 3 that best fits your two-dimensional dataset.

The exponential regression calculator helps you find the exponential curve that best models your dataset.

The Fisher's exact test calculator performs a one-tailed and two-tailed Fisher's test on any given 2 x 2 contingency table.

Use the F-statistic calculator to compare the variances of two given populations.

The linear regression calculator determines the coefficients of linear regression model for any set of data points.

Calculate precise margins of error with our easy-to-use calculator. Get accurate results using the margin of error formula. Try it now!

Our McNemar's test calculator can perform several different versions of this popular statistical test for paired binomial data, including the exact binomial test!

Use the normal approximation calculator to approximate the probability for a binomial event with a normal distribution.

The polynomial regression calculator can help you find a polynomial curve that best fits your data set.

Quadratic regression calculator determines the parabola that best fits any given set of points.

The raw score calculator will help you to determine the original data point from the mean value, standard deviation, and z-score.

Find out what is the relative error and absolute error, and discover the difference between the two, with our relative error calculator!

The residual calculator helps you to calculate the residuals of a linear regression analysis.

Use this tool to calculate the sampling error incurred when inferring from a population.

Use the T statistic calculator to estimate the t-value of a given dataset.

The Mann-Whitney U test calculator performs this popular statistical test using the beloved U statistic.

The Wilcoxon rank-sum test calculator allows you to perform the famous non-parametric test based on the ranks of observations. Both exact and approximated versions are available, along with corrections for ties and continuity!

The sample size calculator determines the sample size required for the result to be statistically significant.

Check out this Shannon entropy calculator to find out how to calculate entropy in information theory.

Use this 10-sided dice roller calculator to roll up to 15 10-sided dice at a time.

Our 2 dice roller calculator simulates a toss of two dice and much more!

Set up to 15 dice to roll with our 4-sided dice roller calculator.

Use 6 sided dice probability calculator to find the likelihood of rolling numerous numbers sets with standard cubic dice.

Our 6-sided dice roller calculator simulates the toss of the most common dice type and much more!

If you don't like statistics, but life is forcing you to count the 90% confidence interval, this calculator will significantly help.

The 95% confidence interval calculator will be your buddy in calculating the range in which you can be 95% sure of your result.

The 99% confidence interval calculator will help you with the fundamental statistics problem of estimating a specific confidence interval.

Learn how to calculate the joint probability of two events with Omni's AND probability calculator.

Sort numbers from least to greatest with this ascending-order calculator.

Use our combinations without repetition calculator to know the possible combinations without repetition from a group of objects.

This tool calculates the combinations with repetition of a set of objects. But be careful! Please don't confuse it with permutation!

This custom dice roller calculator will help you with boredom and roll up to 15 dice with up to 120 sides!

This fast D100 dice roller calculator will allow you to throw up to 15 D100 dice at a time.

Roll online multiple D20 dice for Dungeons & Dragons or any tabletop game with our D20 dice roller calculator.

Use our decimal odds calculator to quickly find your total return in case of a winning bet.

This decimal random number generator can output sequences of uniformly distributed random numbers from any given range.

Use DnD Dice Roller Calculator to easily roll multiple dice for your tabletop game.

The false positive calculator finds the rate of false positives based on the specificity and prevalence.

Our first quartile calculator will teach you all there is about the lower quartile.

Use this fractional odds calculator to find the odds of a situation in fraction form.

Sort numbers in descending order with this greatest-to-least calculator.

Use our least to greatest decimals calculator to sort decimals in ascending order.

Use this tool to calculate the lower fence of a set of numbers.

Find out how much you can win or lose if you wager $100 with the moneyline odds calculator's help.

Use this calculator to order from least to greatest, for any list of numbers.

Use this calculator to order decimals in ascending or descending order

With our ordering numbers calculator, you can quickly rearrange up to 50 numbers in ascending or descending order.

This or that? The OR probability calculator will teach you how to find the probability of one outcome, the other, or both.

Count exactly what the name says with our permutation without repetition calculator!

Find out how many different arrangements of a set of objects are possible using this permutation with repetition calculator.

With this pie chart angle calculator, you will be able to quickly create pretty pie charts!

With the pie chart percentage calculator, you can find the percentage fraction of categories in a data set, and display them in a pie chart.

Use our possible combinations calculator to know how many combinations are possible, given a total number of objects and a sample size.

Learn how to express probability as a fraction with our probability fraction calculator!

A set of x-y data is fitted with a 4th-order polynomial model using the quartic regression calculator.

The random dice roller can roll up to fifteen dice at once and output a list of their results.

This sensitivity calculator will introduce you to this key concept related to the accuracy of diagnostic tests.

With our specificity calculator, you can learn how to examine the accuracy of diagnostic tests.

This third quartile calculator can help you discover all the interesting properties of the upper quartile.

Use the two dice probability calculator to find the likelihood of running different types of two dice.

Efficiently determine limits with our upper and lower fence calculator. Perfect for statistical analysis. Simplify your data evaluation now!

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Teach yourself statistics

Binomial Probability Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution .

• Enter a value in each of the first three text boxes (the unshaded boxes).
• Click the Calculate button to compute binomial and cumulative probabilities.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What is a binomial experiment?

A binomial experiment has the following characteristics:

• The experiment involves repeated trials.
• Each trial has only two possible outcomes - a success or a failure.
• The probability that any trial will result in success is constant.
• All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution . It refers to the probabilities associated with the number of successes in a binomial experiment .

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution , as shown below.

What is the number of trials?

The number of trials refers to the number of replications in a binomial experiment.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. We classify Heads as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

Note: Each trial results in a success or a failure. So the number of trials in a binomial experiment is equal to the number of successes plus the number of failures.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

In a binomial experiment, the probability that the experiment results in exactly x successes is indicated by the following notation: P(X=x);

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X<2); the probability of getting AT MOST 2 successes is indicated by P(X≤2); the probability of getting AT LEAST 2 successes is indicated by P(X≥2); the probability of getting MORE THAN 2 successes is indicated by P(X>2).

Sample Problem

• The probability of success (i.e., getting a Head) on any single trial is 0.5.
• The number of trials is 12.
• The number of successes is 7 (since we define getting a Head as success).

Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button.

The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probabilities. For example, the probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative probability equal to 0.806.)

• The probability of success for any individual student is 0.6.
• The number of trials is 3 (because we have 3 students).
• The number of successes is 2.

The calculator reports that the probability that two or fewer of these three students will graduate is 0.784.

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