A jar contains 30 green jelly beans and 20 purple jelly bean

A jar contains 30 green jelly beans and 20 purple jelly beans.Suppose 10 jelly beans are selected at random from the jar. A) Find the probability of obtaining exactly five purple jellybeans if they are selected with replacement.

B) Find the probability of obtaining exactly five purple jellybeans if they are selected without replacement

Solution

there are all total 50 jelly beans. out of which 30 are green and 20 are purple. 10 jelly beans are selected.

so the probability of selecting a purple jellybean is 20/50=2/5 and probability of not selecting a purple jelly bean is 30/50=3/5

A) let X be the number of purple jellybeans selected with replacement.

so X~Bin(10,2/5)

so P[X=5]=¹⁰C₅*(2/5)⁵(3/5)⁵=0.200658 [answer]

B) let Y be the number of purple jelly beans selected without replacement.

so Y~hypergeometric(50,20,10)

so P[Y=5]=²⁰C₅*^50-20C_10-5/⁵⁰C₁₀=0.215085 [answer]