For every $1 you wager on white at this point, you expect to lose 1/19 of $1, or a little more than 5¢. That is, in the long run your expectation is to lose 5.26°h of your wager. Again, each individual wager will result in either a S 1 win or a S 1 loss, but over the long haul you will lose more often than win, which leads to your demise at an average rate of 5¢ per play.
Now consider a situation where there are 10 white and 9 black gumballs left. In this case, you have the advantage, and it's times like these that you will go ahead and make the bet. Following the above example, it's easy to see that the expectation is +5.26%. Every bet you make in this situation is a long-run moneymaker.

Clearly, if your intention is to play this game for profit, you should play only when you have a positive expectation, and never play when the expectation is negative. So the question is: How can you identify the times when making the bet is favorable? A simple way to determine whether or not you have the advantage is to track the gumballs so you have information about how many of each color remain in the dispenser. We know that removing black gumballs from the machine helps you: Every time a black ball is taken out of play, your expectation (for betting on white) goes up slightly. The opposite is true when a white gumball is removed. So to begin, we can ass -n black balls a value of +l and white balls a value of -l (these assignments are sometimes referred to as "tags").

Start at Zero and keep a "running count" of all the balls as they come out of the machine. After a ball is seen, the running count is updated by adding its value. For example. if the first ball to come out is black, then the running count is + 1. If the next ball is white, the running count goes back to 0 (arrived at by starting with +l for the old running count and adding -1 for the white ball that just came out).

As you may have deduced, the running count alerts you to when it's profitable to play-whenever the running count is positive, you have the advantage. You don't need to count (and remember) the exact number of white and black balls played; you don't even need to know how many balls are left. You need only know the value of the running count to know whether or not you have an edge. The point at which we know you first have the advantage is called the "key count." At or above the key count (+l), you have the advantage; below the key count, you are either neutral or at a disadvantage.