Expected Value w/o replacement

Question

Jack and Jill each have a bag of balls numbered 1 through 25. Jack draws 17 balls without replacement from his bag and Jill draws 20 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

I am not 100% certain of this answer, but I will give you something to think about.

 

Jack makes a selection of 17 numbers out of 25 and leaves 8; that divides the numbers into 2 groups.  Now Jill gets to draw.

 

 

Then Jill can take the following pairs: (17,3), (16,4), (15,5), (14,6), (13,7), (12,8) from each of the 2 groups left by Jack & the first number in the pair is the number of matches.

 

She can make these picks in the following ways, e.g. for the first pair ₁₇C₁₇ * ₈C₃ and similarly for the remaining possibilities.

 

The probability of each pair is determined by dividing the number of ways of selecting each pair divided by ₂₅C₂₀.

 

The expected number of matches is obtained by multiplying the probability of each pair by the first number in the pair and adding those products for each of the 6 possibilities.

 

BTW: _nC_r is the binomial coefficient of n things taken r at a time.  The arithmetic is a nuisance at best, but if you use a spreadsheet to make the table of binomial coefficients up to 17, then it will not be too bad.

 

Now as I said when I started I am not 100% sure this is correct.  If you get a better and simpler answer, please let me know...or if your instructor gives you a better answer, I would appreciate knowing that too.