Suppose 7 of 36 bottles of wine are bad If you randomly sele

Suppose 7 of 36 bottles of wine are bad. If you randomly select 5

bottles without replacement, find the probability that:

a) Exactly 3 of the bottles are bad.

b) At least 3 of the bottles are bad.

c) All 5 bottles are good or all 5 bottles are bad.

Please include steps with the answer

Solution

Since this is sampling without replacement we use hypergeomentric distribution.

a: Number of bad bottles in the population

n: total number of bottles

a=7

n=36

b=n-a=29

a) P(exactly 3 bottles are empty)

x: number of bd bottles.

P(x=3)=0.037693

This is calculated using HYPGEOM.DIST(3,5,7,36,FALSE) function in excel.

b) P(atleast 3 bottles are bad)=P(x>=3)=P(x=3)+P(x=4)+P(x=5)

Totalling we get P(x>=3)=0.04044

c) P(all 5 bottles are good or all 5 bottles are bad)

P(x=0)+P(x=5)

all bottles are good then x=0, since x is the number of bad bottles.

Totalling we get 0.315062.