Question

Let's consider an urn that contains 15 balls, of which 5 are black balls. An integer n is randomly selected from the set {1, 2, 3, 4, 5, 6, 7, 87}, and then a sample of size n is obtained without replacement of the urn. Find the probability that all the balls in the sample are black.

Answer 1

Probability of selecting a random integer from the set A={1,2,3,4,5,6,7,8} =1/8

First case, let n=1, that means if we take a sample of size 1 obtained without replacement, then

Probability that the selected ball is black =number of black balls/ total number of balls=5/15

Case 2, let n=2, that means we take a sample of size 2 without replacement

Probability that all the selected balls (here 2 balls) without replacement from the urn are black= Probability of first ball being black X probability of second ball being black = (5/15)(4/14) (using fundamental principle of counting)

(Probability of choosing second ball to be black after we get first black ball without replacement = number of black balls left/total number of balls left = 4/14)

Case 3, let n=3, sample of size 3

Probability that all the selected balls (here 3 balls) without replacement from the urn are black= (5/15)(4/14)(3/13)

Continuing in this manner for n=4 and n=5,

Probability that all the 4 selected balls without replacement from the urn are black= (5/15)(4/14)(3/13)(2/12)

Probability that all the 5 selected balls without replacement from the urn are black= (5/15)(4/14)(3/13)(2/12)(1/11)

For n>5,

Now since there are only 5 black balls in the urn, so probability that all the balls in the sample of more than 5 balls are black = 0

Probability that all the balls in the sample are black=(Probability of choosing a random integer from the set A)*(Probability that all the balls in the sample of size 1 are black+Probability that all the balls in the sample of size 2 are black+ Probability that all the balls in the sample of size 3 are black +-------+Probability that all the balls in the sample of size 8 are black)

=(1/8)[(5/15)+(5/15)(4/14)+(5/15)(4/14)(3/13)+(5/15)(4/14)(3/13)(2/12)+(5/15)(4/14)(3/13)(2/12)(1/15)+0+0+0]

=(1/8)(0.4543) =0.056 Ans