Hypergeometric Distribution Probability Calculator (2024)

The Hypergeometric Calculator makes it easy to compute individual and cumulative hypergeometric probabilities. For help, read the Frequently-Asked Questions or review the Sample Problems.

To learn more, read Stat Trek'stutorial on the hypergeometric distribution.

Frequently-Asked Questions


Calculator | Sample Problems

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

What is a hypergeometric experiment?

A hypergeometric experiment has two distinguishing characteristics:

  • The researcher randomly selects, without replacement, a subset of items from a finite population.
  • Each item in the population can be classified as a success or a failure.

Suppose, for example, that we randomly select 5 cards from an ordinary deck of playing cards. We might ask: What is the probability of selecting exactly 3 red cards? In this example, selecting a red card (a heart or a diamond) would be classified as a success; and selecting a black card (a club or a spade) would be classified as a failure.

What is a hypergeometric distribution?

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

For example, suppose we randomly select 5 cards from an ordinary deck of playing cards. We might ask: What is the probability distribution for the number of red cards in our selection. In this example, selecting a red card would be classified as a success. The probabilities associated with each possible outcome are an example of a hypergeometric distribution, as shown below.

Outcome Hypergeometric Prob Cumulative Prob
0 red cards 0.025 0.025
1 red card 0.150 0.175
2 red cards 0.325 0.500
3 red cards 0.325 0.825
4 red cards 0.150 0.975
5 red cards 0.025 1.00

Given this probability distribution, you can tell at a glance the individual and cumulative probabilities associated with any outcome. For example, the individual probability of selecting exactly one red card would be 0.15; and the cumulative probability of selecting 1 or fewer red cards would be 0.175.

What is a population size?

In a hypergeometric experiment, a set of items are randomly selected from a finite population. The total number of items in the population is the population size.

For example, suppose 5 cards are selected from an ordinary deck of playing cards. Here, the population size is the total number of cards from which the selection is made. Since an ordinary deck consists of 52 cards, the population size would be 52.

What is a sample size?

In a hypergeometric experiment, a set of items are randomly selected from a finite population. The total number of items selected from the population is the sample size.

For example, suppose 5 cards are selected from an ordinary deck of playing cards. Here, the sample size is the total number of cards selected. Thus, the sample size would be 5.

What is the number of successes?

In a hypergeometric experiment, each element in the population can be classified as a success or a failure. The number of successes is a count of the successes in a particular grouping. Thus, the number of successes in the sample is a count of successes in the sample; and the number of successes in the population is a count of successes in the population.

What is a hypergeometric probability?

A hypergeometric probability refers to a probability associated with a hypergeometric experiment. For example, suppose we randomly select 5 cards from an ordinary deck of playing cards. We might ask: What is the probability of selecting EXACTLY 3 red cards? The probability of getting EXACTLY 3 red cards would be an example of a hypergeometric probability, which is indicated by the following notation: P(X = 3).

The probability of getting exactly 3 red cards is 0.325. Thus, P(X = 3) = 0.325. (The probability distribution showing this result can be seen above in the question: What is a hypergeometric distribution?)

What is a cumulative hypergeometric probability?

A cumulative hypergeometric probability refers to a sum of probabilities associated with a hypergeometric experiment. To compute a cumulative hypergeometric probability, we may need to add one or more individual probabilities.

For example, suppose we randomly select 5 cards from an ordinary deck of playing card. We might ask: What is the probability of selecting AT MOST 2 red cards? The cumulative probability of getting AT MOST 2 red cards would be equal to the probability of selecting 0 red cards plus the probability of selecting 1 red card plus the probability of selecting 2 red cards. Notationally, this probability would be indicated by P(X < 2).

The cumulative probability for getting at most 2 red cards in a random deal of 5 cards is 0.500. Thus, P(X < 2) = 0.500. (The probability distribution showing this result can be seen above in the question: What is a hypergeometric distribution?)

Sample Problems


Calculator | Frequently-Asked Questions


  1. Suppose you select randomly select 12 cards without replacement from an ordinary deck of playing cards. What is the probability that EXACTLY 7 of those cards will be black (i.e., either a club or spade)?

    Solution:

    We know the following:

    • The total population size is 52 (since there are 52 cards in the deck).
    • The total sample size is 12 (since we are selecting 12 cards).
    • The number of successes in the population is 26. (Here, we define a success as choosing a black card, and there are 26 black cards in an ordinary deck of playing cards.).
    • The number of successes in the sample is 7 (since there are 7 black cards in the sample that we select).

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

    Hypergeometric Distribution Probability Calculator (1) Hypergeometric Distribution Probability Calculator (2)

    The calculator reports that the hypergeometric probability is 0.20966. That is the probability of getting EXACTLY 7 black cards in our randomly-selected sample of 12 cards.

    The calculator also reports cumulative probabilities. For example, the probability of getting AT MOST 7 black cards in our sample is 0.83808. That is, P(X < 7) = 0.83808.
  1. Suppose we are playing 5-card stud with honest players using a fair deck. What is the probability that you will be dealt AT MOST 2 aces? (Note: In 5-card stud, each player is dealt 5 cards.)

    Solution:

    We know the following:

    • The total population size is 52 (since there are 52 cards in the full deck).
    • The total sample size is 5 (since we are dealt 5 cards).
    • The number of successes in the population is 4 (since there are 4 aces in a full deck of cards).
    • The number of successes in the sample is 2 (since we are dealt 2 aces, at most.).

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

    Hypergeometric Distribution Probability Calculator (3) Hypergeometric Distribution Probability Calculator (4)

    The calculator reports that the P(X < 2) is 0.99825. That is the probability we are dealt AT MOST 2 aces. The cumulative probability is the sum of three probabilities: the probability that we have zero aces, the probability that we have 1 ace, and the probability that we have 2 aces.

Hypergeometric Distribution Probability Calculator (2024)

FAQs

How do you find the probability of a hypergeometric distribution? ›

The formula for the hypergeometric probability distribution is f(x) = (k x)(n-k n-x)/(N n). N is the size of the population being sampled, n is the size of the sample, and k is the number of "successes" in the population.

How to calculate hypergeometric p value? ›

In a test for over-representation of successes in the sample, the hypergeometric p-value is calculated as the probability of randomly drawing k or more successes from the population in n total draws. In a test for under-representation, the p-value is the probability of randomly drawing k or fewer successes.

What is the CDF of a hypergeometric distribution? ›

The CDF function for the hypergeometric distribution returns the probability that an observation from an extended hypergeometric distribution, with population size N, number of items R, sample size n, and odds ratio o, is less than or equal to x.

What is probability mass function of hypergeometric distribution? ›

then the probability mass function of the discrete random variable is called the hypergeometric distribution and is of the form: P ( X = x ) = f ( x ) = ( m x ) ( N − m n − x ) ( N n )

What is the formula for the hypergeometric equation? ›

z ⁢ ( 1 − z ) ⁢ d 2 w d z 2 + ( c − ( a + b + 1 ) ⁢ z ) ⁢ d w d z − a ⁢ b ⁢ w = 0 . This is the hypergeometric differential equation. It has regular singularities at z = 0 , 1 , ∞ , with corresponding exponent pairs { 0 , 1 − c } , { 0 , c − a − b } , { a , b } , respectively.

What is the probability distribution formula? ›

Probability Distribution Function

It can be written as F(x) = P (X ≤ x). Furthermore, if there is a semi-closed interval given by (a, b] then the probability distribution function is given by the formula P(a < X ≤ b) = F(b) - F(a).

What is the expected value of a hypergeometric distribution? ›

Expected Value: The expected value, also called the mean, of a hypergeometric distribution is the predicted number of successes in an experiment.

What is the notation for the hypergeometric probability distribution? ›

X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.

How do you find the p-value of a distribution? ›

The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test). The p-value for: a lower-tailed test is specified by: p-value = P(TS ts | H 0 is true) = cdf(ts)

What is the difference between hypergeometric and binomial distribution? ›

For the binomial distribution, the probability is the same for every trial. For the hypergeometric distribution, each trial changes the probability for each subsequent trial because there is no replacement.

Is hypergeometric distribution a continuous probability distribution? ›

The hypergeometric distribution is an example of a discrete probability distribution because there is no possibility of partial success, that is, there can be no poker hands with 2 1/2 aces. Said another way, a discrete random variable has to be a whole, or counting, number only.

What is the approximation of a hypergeometric distribution? ›

Still, in practice, a hypergeometric distribution can usually be approximated by a binomial distribution. The reason for this is that if the sample size does not exceed 5 % of the population size, there is little difference between sampling with and without replacement.

How to calculate hypergeometric probability? ›

If a random experiment satisfies all of the above, the distribution of the random variable X, where X counts the number of successes, is called a hypergeometric distribution, we write X ~ H(n, a, N). The hypergeometric distribution is P(X = x) = aCx⋅bCn−xNCn, x = 0, 1, 2, … , n or a, whichever is smaller.

What does c mean in hypergeometric distribution? ›

Cx: The number of combinations of k things, taken x at a time. h(x; N, n, k): hypergeometric probability - the probability that an n-trial hypergeometric experiment results in exactly x successes, when the population consists of N items, k of which are classified as successes.

What is the formula for probability? ›

Probability determines the likelihood of an event occurring: P(A) = f / N. Odds and probability are related but odds depend on the probability. You first need probability before determining the odds of an event occurring. The probability types are classical, empirical, subjective and axiomatic.

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