What is my expected winnings?

Question

A box contains 7 black blocks, numbered 1-7, and 7 yellow blocks, numbered 1-7. You randomly select two blocks, without replacement without regard to color. The absolute difference of the numbers on the two blocks selected is how much you win. What is my expected winnings?

I’m not sure of how to solve. I started using a tree diagram which got lengthy extremely fast. Can i use binomoial distribution to solve this using p=8 x=2 n=16?

Answer 1

The only way I know to solve this problem is to make a table of the sample space.

But first: the probability of 2 numbers the same is (1/7)(1/13) and the probability of unmatched pairs is (1/7)(2/13).

Now make a table with 1-7 down the side and 1-7 across the top; at the intersection of each row and column write the absolute value of the difference. Count the occurrences of each of the possible outcomes. There are 7 ways of matched pairs for probability 1/13. There are 12 ways to get 1 for probability 24/91...and so on.

Once you have the list of probabilities for each absolute value, then the expectation is simply the sum of the probabilities multiplied by the value (0 thru 6).

If I could see a short cut, I would tell you...but I don't see one.