You look at a carnival game. The person running it says, "Just reach your hand into this bag. There are 9 yellow balls and 1 red ball in the bag. You get 4 chances to pull out the red ball. (You have to put the ball you drew back before you draw another ball.) You only have to pay one dollar to play, and you get 3 dollars if you pull out the red ball!"

Assuming the person running the game is telling the truth, and the balls only differ in color, would you expect to make a net profit or a net loss on this game?

(In reply to solution by Bart freeman)

Bart: In order to help you see the difference between your solution and the solution most of the others have come up with, I am going to simplify the conditions so that all possible outcomes can be listed, and the probability shown directly.

Assume that there are only two balls, one red and one yellow, and that you have two chances to find the red. By your reasoning, you have a 50% chance of winning. By Fwaff's and the others' reasoning you have a 1-(50%)² = 75% chance of winning.

Now during the first draw, there are two possibilities with equal chance of happening: You can draw the red ball, or you can draw the yellow ball. On your second draw, you have the same two possibilities with equal chances.
RR = win (25%)
RY = win (25%)
YR = win (25%)
YY = lose (25%).

I think where you went wrong, is that you forgot that you paid for two (or in the original problem, four) chances to draw. The fact that after a victory is assured you do not take your remaining draws does not change the odds. They could be restated:
Rx = win (50%)
YR = win (25%)
YY = lose (25%).

Posted by TomM on 2003-09-20 21:10:23