Bobby lives in the country and is visited by his two chess-playing cousins
from the big city.  His cousins are single and would like to meet two women
who also like to play chess.
     Bobby knows a family with many adult sisters, some who like to play chess
in the park and some who don't play chess but walk in the park.
     Bobby instructs his cousins to go to the park and if they wait long enough
to meet two of the sisters then there is a 50-50 probability that both sisters will
be chess players.
     How many sisters are there total?

There are 4 sisters, 3 of them are chessplayers.

If Bobby's cousins try to figure this out, they are doomed to a 0% probability of actually meeting the chess chicks.   :)

rooperi,

How did you arrive at that answer?

 seems right. if there are 4 sisters A, B, C, D and B, C and D play chess, then you could meet the following combinations: AB, AC, AD, BC, BD, CD. The last 3 comprise 50% of your chances of encountering two chess players.

What he said *points up*

 the credit is all yours. i just tested your answer to see if it's right.

Let's see...the triangle numbers include:

0: 0
1: 1
2: 3
3: 6
4: 10
5: 15
6: 21
7: 28
8: 36
9: 45
10: 55
11: 66
12: 78
13: 91
14: 105
15: 120

The only numbers that meet the criteria are triangle numbers 2 and 3, so there are 3+1 sisters with 2+1 of them being chess players, or:

4 sisters, 3 are chess players

Are you all sure this is the correct answer?