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?

Much of the reasoning given previously is correct- given a single play of the game, which involves UP TO 4 draws, you have a 34.39 percent chance of winning 2 dollars (3 dollar prize minus 1 dollar entry fee) and a 65.61 percent chance of losing 1 dollar. What this means for the problem, however, depends on how we disambiguate the wording of the question. On the one hand, if the question is meant to be read as:

[1] On a single play of the game (i.e. up to 4 draws), would you expect to make a profit?

then the answer is of course no, since the chances of winning (regardless of how much you might win or lose) is quite a bit less than 50 percent. However, if the question is meant to be read as:

[2] At the limit, i.e. on average, should one expect to make a profit from this game if one plays long enough?

or even better:

[3] Should the carnival, in setting up this game, expect to report a loss regarding it on their tax forms at the end of the year (assuming no tax fraud)?

then the answer is yes, since on average the game will pay out more than it collects (the 2 dollar prize being won 34.39 percent of the time outweighing the 1 dollar fee being taken 65.61 percent of the time, for reasons cited by others earlier).

Posted by RoyCook on 2003-09-19 13:20:22