Hypergeometric Distributions

Remember, for rolling dice

uniform

Rolling a 4

P(4) = 1/6

binomial

Rolling a 7

P(pair) = 6/36

geometricRolling a pairP(pair) = eventually

A Hypergeometric distribution does not fit any of these

models.

It is close to the Binomial Model.

HD are used when the probabilities of successive trials (or selections)

are dependent

Meeting the first condition determines the probability of the second.

The random variable (X) is the number of successful trials in an experiment.For efficiency, we will use combinations to help us count

Jury SelectionA six member jury is needed.

There are 10 women and 6 men available.

Create a probability distribution for all the possible juries.

Identify the DRV X as the number of women on the jury.

Try a binomial model….

Jury SelectionA six member jury is needed.

There are 10 women and 6 men available.

Create a probability distribution for all the possible juries.

X: a womanp = 10 / 16q = 6 /16

Suppose we select 4 women, so x = 4There is a breakdown when we model this…..not the same as dice….not all the probabilities stay the same

Jury SelectionA six member jury is needed.

There are 10 women and 6 men available.

Create a probability distribution for all the possible juries.

X: a womanp = 10 / 16q = 6 /16

Suppose we select 4 women, so x = 4Our way around this is to chose the women in groups (like subsets, we have done this before….)

Determine the PD for the number of women selected for a 6 member jury

from a group of10 women and 8 men

Let (the RV) X be the number of women selected.

Number of women (x) Probability P(x)

P(x) = n(x)

n(S)

= women X men

Total number of

selections

= 10Cx X 8C6 - x

18C6

Probability in a Hypergeometric Distribution

P(x) = aCx X n-aCr-x

nCrx: number of successful selections

r: size of the subset drawn (6)

a: total number of successful elements to draw from (10)

n: all selections (18)

Expected ValueE(X) = r a

nWhat is the expected number of women on the committee?

r = 6

a = 10

n = 18E(X) = 6[10/18]

= 3.3

Homework

Pg 404 1,3,7,11,12

Remember

1. If you roll a die 4 times, what is the probability that

a) you roll a 3 twice?

b) you roll a 3 on the last roll?