Question

Jack and Jill each have a bag of balls numbered 1 through 31. Jack draws 15 balls without replacement from his bag and Jill draws 12 balls without replacement from her bag. If they both draw the same numbered ball they call it a match. What is the expected number of matches?

Answer 1

Ans Let us first calculate in how many ways can Jack take out 15 balls without replacement which will be basically 31C15 denoted by K since it is basically taking 15 balls out of 31 balls without replacement and arrangement of which doesn't matter.

Similarly for Jill the number of ways for taking out 12 balls is 31C12 which we denote by S

Now lets calculate the probability of match of one ball for which we first calculate the number of ways it is possible.

We can select the number to matched in 31C1 ways.Selecting that ball Jack can take the rest 14 balls in 30C14 ways and Jill now will have 16 options out of which she can take 11 balls in 16C11 ways.

So the required probability is (31C1 * 30C14 * 16C11) / (K*S)

Similarly we can calculate the probability for 2 matches in following way

We can select the 2 numbers to matched in 31C2 ways.Selecting that ball Jack can take the rest 13 balls in 29C13 ways and Jill now will have 16 options out of which she can take 10 balls in 16C10 ways.

So the required probability is (31C2 * 29C13 * 16C10) / (K*S)

Similarly we can calculate the probability for all values from 3 to 12 since the maximum number of matches will be 12.

So the required expected value is

since Expected value is Summation of Value * Probability of getting that value for all the values which is here from 0 to 12

So on calculating the Expected value we will get 5.8063 approximately.