A deck of 2n cards consists of n red and n black cards The c

A deck of 2n cards consists of n red and n black cards. The cards are shuffled and then turned over one at a time. Suppose that each time a red card is turned over, we win 1 unit if more red cards than black cards have been turned over by that time. (For instance, if n = 2 and the result is r b r b, then we would win a total of 2 units.) Find the expected amount that we win.

Solution

There are equal red and and black cards.

Thus from symmetry,
in any given turn there is equal probability that red will outnumber black so far,
or black will outnumber red.

Thus, if we find the possible deals where red and black are equal by any one turn (Nequal),
we can subtract it from the total number of possible deals (2^n) and divide by two to get the number of deals where red > black at this turn (Nr).

Then the expected number to be added at this turn is Nr/2^n