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Here is one stopping rule that guar- antees winning more than half the time. First, generate a random number R ac- cording to a standard Gaussian (bell- shaped) curve by using a computer or other device. Then turn over one of the slips of paper and observe its number. If R is larger than the observed number, continue and turn over the second card. If R is smaller, quit with the number ob- served on the first card. How can such a simple-minded strategy guarantee a win more than half the time? If R is smaller than each of the two written numbers, then you win exact- ly half the time (p/2 of the unknown probability p in Figure 4); if it is larger than both, you again win half that time (q/2 of q, also in Figure 4). But if R falls between the two written numbers, which it must do with strictly positive probability (since the two numbers are different and the Gaussian distribu- tion assigns positive probability to ev- ery interval) then you win all the time. This gives you the edge you need, since p/2 + q/2 + 1–p-q is greater than 1/2, because 1-p-q is greater than zero. For example, if the two hidden num- bers are 1 and π, this Gaussian method yields a value for p about .8413 and q about .0008, so the probability that it will select the larger number is more than 57 percent.

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