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Wize University Statistics Textbook > Discrete Probability Distributions

Hypergeometric Probability

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Hypergeometric Experiment


A hypergeometric experiment has the following properties:

  1. You are randomly drawing a sample of size nn (fixed number of trials) from a population of size NN without replacement
  • i.e. each draw is not independent
  • i.e. the probability of success is not constant from trial to trial
  1. rr of the NN objects in the population are considered "success" (a.k.a. favorable)

Watch Out!
"Success" just denotes the outcome that you are interested in.
"Failure" just denotes the opposite outcome (i.e. complement).

For example, these could be "male or female", "win or lose", "greater than 1 or not greater than 1", etc.

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Examples
Determine if the following are hypergeometric experiments:

1. You draw coins from a bag containing 3 red coins and 5 blue coins and note the colour of the coin drawn. You repeat this experiment 4 times with replacement →
NO

This is NOT hypergeometric since each trial is independent from the other ones because you are doing this with replacement (the bag starts off with the same coins at the start of every trial.

2. You draw coins from a bag containing 3 red coins and 5 blue coins and note the colour of the coin drawn. You repeat this experiment 4 times without replacement →
YES

This is hypergeometric since each coin can either be success - red or failure -blue (you could switch these labels as well), there's a fixed number of draws from a total population of objects with replacement (each draw is not independent)
  • N=8N=8 (total number of coins)
  • r=3r=3 if red is favorable; r=5r=5 if blue is favorable (the number of objects considered as "successes")
  • n=4n=4 (the number of draws or trials)
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Hypergeometric Distribution

Let XX be a random variable that represents the number of successes in a hypergeometric experiment where
  • N=N= number of all available items (population size)
  • n=n= number of trials or number of items drawn from NN (sample size)
  • r=r= number of favorable items available in NN (successes)
  • x=0,1,2,3...x=0,1,2,3...
Then XX follows a hypergeometric distribution (or hypergeometric model), and we denote this by X∼Hypergeometric(N,r,n)\boxed{X\sim \text{Hypergeometric}(N,r,n)} or X∼H(N,r,n)\boxed{X\sim H(N, r, n)}.


Then the probability of obtaining exactly xx number of successes is given by this formula:
P(X=x)=(rx)(N−rn−x)(Nn)\displaystyle{\boxed{\displaystyle P(X=x)= \displaystyle \frac{{r \choose x}{{N-r}\choose{n-x}}}{N\choose n}}}
Notes:
  • xx must be less than or equal to the smaller of the numbers nn and rr
  • i.e. you can't have more "successes" than there are number of "success" objects/favorable objects
  • i.e. you can't have more "successes" than the number of objects drawn
x≤min(n,r)x\leq min(n,r)

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Mean, Variance, and Standard Deviation

  • Mean: E(X)=n×rN\boxed{E(X)=n\times\frac{r}{N}}
  • Variance: Var(X)=σ2=(n).(rN).(1−rN).(N−nN−1)\boxed{Var(X)=\sigma^2=\left(n\right).\left(\frac{r}{N}\right).\left(1-\frac{r}{N}\right).\left(\frac{N-n}{N-1}\right)}
  • Standard deviation: SD(X)=σ=(n).(rN).(1−rN).(N−nN−1)\boxed{SD(X)=\sigma=\sqrt{\left(n\right).\left(\frac{r}{N}\right).\left(1-\frac{r}{N}\right).\left(\frac{N-n}{N-1}\right)}}
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Practice: Hypergeometric Distribution

There are seven people in Molly’s family -- 4 women and 3 men. If I select three of them at random, what is the probability that I will get exactly two men and one woman?

Let men be "success"

x=2N=7n=3r=3\begin{array}{c}x=2\\N=7\\n=3\\r=3\end{array}

P(X=2)\displaystyle P(X\bm=2)

=(xr)(n−xN−r)(nN)\displaystyle =\frac{(^r_x)(^{N-r}_{n-x})}{(^N_n)}

=(23)(3−27−3)(37)\displaystyle =\frac{(^3_2)(^{7-3}_{3-2})}{(^{7}_3)}

=(3)(4)35\displaystyle =\frac{\left(3\right)\left(4\right)}{35}

=1235\displaystyle =\frac{12}{35}

≈0.3429\approx 0.3429

Alternative method - Let women be "success"

x=1N=7n=3r=4\begin{array}{c}x=1\\N=7\\n=3\\r=4\end{array}

P(X=1)\displaystyle P(X\bm=1)

=(xr)(n−xN−r)(nN)\displaystyle =\frac{(^r_x)(^{N-r}_{n-x})}{(^N_n)}

=(13)(3−17−4)(37)\displaystyle =\frac{(^3_1)(^{7-4}_{3-1})}{(^{7}_3)}

=(4)(3)35\displaystyle =\frac{\left(4\right)\left(3\right)}{35}

=1235\displaystyle =\frac{12}{35}

≈0.3429\approx 0.3429

Same answer!

There are 10 balls in a bag; 3 of them are gold balls. To win a prize, you must draw all three gold balls. You get to draw five balls. What is the probability you will win a prize?

Practice: Hypergeometric Distribution

Ivan supervises employees at Walmart. Interestingly, 15 of his night shift employees called in sick today. Ivan believes that some of them are lying about their sickness to go to the Rhianna concert tonight.

An employee knows that 6 of these 15 employees have Rhianna concert tickets and are not sick – but Ivan doesn’t know this! Ivan hires a detective to randomly spy on 5 of his employees. What is the probability that exactly 2 of the 5 selected employees will be caught with Rhianna tickets?


Extra Practice
Hypergeometric probability
In a box there are 5 gold, 3 silver, and 2 bronze balls. Three balls are drawn without replacement.
Determine the probability that at least one ball selected is silver.
Hypergeometric Distribution
There are seven people in Molly’s family -- 4 women and 3 men. If I select three of them at random, what is the probability that I will get exactly two men and one woman?
Hypergeometric Probability
The farm has 15 chickens, 7 cows, and 3 pigs. Which of the following is based on a hypergeometric distribution?
Binomial Distribution vs. Poisson Distribution vs. Hypergeometric Probability
Binomial [B], Poisson [P], or Hypergeometric [H]? Enter B, P, or H in the boxes below.
Hypergeometric Probability
Twin Pines is a small neighborhood with 157 houses. A “success” is when a home has the OWN channel, and the probability of success is known. A sample is drawn without replacement. Which model is most the appropriate?
Hypergeometric probability
There are 10 keys in front of you. One of the keys will open the prize car. If you get to pick three keys, what is the probability that one of them is the correct key?
Hypergeometric Probability
Ivan supervises night sift employees at Walmart. The Swine Flu is back and 15 of his night shift employees called in sick today. Ivan finds it odd that they all have Swine Flu and believes that some of them may be lying about their sickness to go to the Taylor Swift concert tonight.
An employee knows that of these 15 employees that called in sick, 6 have Taylor Swift concert tickets – but Ivan doesn’t know this! Ivan hires a private detective to randomly spy on 5 of his employees that called in sick. What is the probability that exactly 2 of the 5 selected employees will be caught with Taylor Swift tickets?
Statistical Properties: Practice
True or false? Enter "T" for true; "F" for false (without quotations).
Hypergeometric Probability
There are 8 buttons in front of you. Three of the buttons will set off the alarm. What is the probability that you will not set off the alarm after pressing 3 buttons?
Hypergeometric Probability Question
There are seven postdoctoral research fellows in Dr. Lemon’s lab – 4 women and 3 men. To be fair, she will select four of them at random to accompany her at a convention. What is the probability that she will get exactly two of each gender?
Hypergeometric Probability
There are 10 balls in a bag; 3 of them are gold balls. To win a prize, you must draw all three gold balls. You get to draw five balls. What is the probability you will win a prize?
Hypergeometric Probability
There are 6 pairs of socks in the dryer. Each pair has a different pattern. In other words, only one sock matches with another sock. You are in a hurry to class and randomly grabbed 4 individual socks from the dryer. What is the probability that all four socks have different patterns? (Relatable, right?)