Hypergeometric Distribution Probability Calculator
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's lesson on the hypergeometric distribution.
- Select a probability from the dropdown box.
- Enter a value in each unshaded textbox.
- Click Calculate to compute probabilities.
| Probability | |
| Population size | |
| Number of successes in population | |
| Sample size | |
| Number of successes in sample | |
| Hypergeometric probability: P(X=x) | |
| Cumulative probability: P(X<x) | |
| Cumulative probability: P(X≤x) | |
| Cumulative probability: P(X>x) | |
| Cumulative probability: P(X≥x) | |
| Mean | |
| Standard deviation |
Frequently-Asked Questions
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.
The researcher conducts the experiment and counts the number of successes. Each time the researcher conducts the experiment, the number of successes may change, based on chance. The hypergeometric calculator computes the probability associated experimental outcomes.
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. Using the hypergeometric calculator, you can compute this probability.
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.
How do you compute hypergeometric probability?
This calculator uses the hypergeometric formula to compute hypergeometric probability:
Hypergeometric Formula.. Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items, x of which are successes. Then the hypergeometric probability is:
h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
where nCr is the number of combinations of n things, taken r at a time.
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?)
What is the mean and standard deviation of a hypergeometric distribution?
The mean (μ) and standard deviation (σ) of a hypergeometric distribution are calculated using population size (N), number of successes in the population (k), and sample size (n).
The mean is the average number of successes you'd expect over many repetitions of a hypergeometric experiment.
μ = n * k / N
And the standard deviation measures the typical distance of the number of successes from the mean.
σ = sqrt { n * (k/N) * [ (N - k)/N ] * [ (N - n) / (N - 1) ] }
Sample Problems
-
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.
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.
-
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.
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.
-
Suppose we draw five cards randomly without replacement from a 52-card deck. What is the probability that our five card include at least
one spade and not more than three spades?
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 draw 5 cards).
- The number of successes in the population is 13 (since there are 13 spades in a standard deck of playing cards).
- The minimum number of successes in the sample is 1 (since the five cards we draw must include at least 1 spade.).
- The minimum number of successes in the sample is 3 (since the five cards we draw must include at most 3 spades.).
In the Probability dropdown box, we select P(x1 ≤ X ≤ x2). The, we plug these numbers into the Hypergeometric Calculator: 52 for population size, 13 for number of successes in the population, 5 for sample size, 1 for minimum number of successes, and 3 for maximum number of successes.
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.