Let’s play a game. I pick a number — any number at all (whole number, fraction, whatever). I then flip a coin to decide whether to add 10 or subtract 10 from it, and I tell you the result. You win the game if you can tell me what my original number was.

For example, suppose the number I picked was 31. I flip a coin, getting heads, and therefore add 10, getting 41, and that’s the number I tell you. Now you know that I picked either 31 or 51. It seems that you have no way of deciding between those, so your chance of winning the game is 1 in 2.

I contend that you can win the game with a probability of greater than 1 in 2, although I will admit that I can’t tell you how much greater the probability is.

Let’s introduce a bit of terminology. We’ll call the first number that I pick X. This is the number that you’ve got to guess. Then I flip a coin and add or subtract 10 from it, getting Y, which is the number I tell you. So we know that Y = X + 10 or Y = X – 10.

The secret to you winning the game is to pick your own number, Z, before I tell you Y. It doesn’t matter how you pick Z, so let’s just say that you have a computer program that generates a random number for you (from the standard normal distribution, if you want to get picky).

Now you play the following strategy:

  • If Y < Z then you guess that X = Y + 10 (ie I subtracted 10)
  • If Y > Z then you guess that X = Y – 10 (ie I added 10)

I claim that this strategy will win you the game more than half the time. To see why, we have to consider three cases:

  1. If X < Z – 10 then we always have Y < Z, so you guess X = Y + 10 and get it right 1 time in 2.
  2. If X > Z + 10 then we have Y > Z, so you guess X = Y – 10 and get it right 1 time in 2.
  3. But if X is in the interval [Z – 10, Z + 10] then your guess will always be right. If X = Y – 10 then we always have Y > Z, so you guess X = Y – 10, which is right. And if X = Y + 10 then we have Y < Z, so you guess X = Y + 10, which is right.

As long as there is some chance that X is in this interval, then there is a greater than 50% chance of you winning the game.

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