Question

6. Tom removes the name tags from 14 Christmas gifts of which 6 were for Mrs. Katie and 8 were for Mr. Morris. If the presents are distributed to the two teachers without reopening the packages

a) What is the probability that Mrs. Katie gets 4 of the presents originally intended for her?

b) What is the expected number of correct presents that will be distributed to Mrs. Katie?

Answer 1

The given problem is solved using hypergeometric Distribution

Suppose a population consists of N items, k of which are successes. When n items are drawn Probability of drawing x : successes.

Hypergeometric distribution can be defined as discrete probability distribution describing the probability of getting x successes in n draws without replacement where the sample population is N. Each draw is either success or failure and the population consists of exactly K successes.

Mean of hyper geometric distribution

:

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Total number of Christmas gifts : N= 14

Number of gifts intended for Mrs.Katie :k = 6

Number of gifts intended for Mr. Morris : = 8

Number of gifts distributed to Mrs. Katie : n=6

X : Number of presents Mrs. Katie will get that are originally intended for Mrs.Kaite

Success : Katie getting a present intended for her

Failure : Katie not getting a present that is not intended for her

X is Hyper geometric distribution and the probability mass function of X

Probability that Mrs.Kaite will get 'x' gifts that intended for her =

Probability that Mrs. Katie gets 4 of the presents originally intended for her = P(X=4)

Probability that Mrs. Katie gets 4 of the presents originally intended for her = 420/3003 = 0.13986014

b) expected number of correct presents that will be distributed to Mrs. Katie

expected number of correct presents that will be distributed to Mrs. Katie = 2.571428571