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Guess the biggest

Master playsLevel 3/5

It belongs to the same constellation as the Game of Googol and the secretary problem: deciding better than at random when you only see part of the information.

Two different numbers between 0 and 1 are chosen at random. You see only one of them.

Then, without knowing the other one, you must decide if the hidden number is greater or less than the one you have seen. At first glance, it seems impossible to do better than chance.

Is there a strategy that maximizes your probability of success? And, if it exists, what is that probability?

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Hints

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  1. If you see a very small number, it seems reasonable to think that the hidden number is larger; If you see a very large one, the opposite.
  2. The probability that the other number is greater than x is 1-x.
  3. It is advisable to change criteria exactly at the point where both options are equally good.

Solution

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Suppose the visible number is x. So:

  • the probability that the hidden number is greater is 1-x
  • and the probability that it is smaller is x Therefore:
  • if x < 1/2, it is convenient to say “the hidden one is greater”
  • if x > 1/2, it is convenient to say “the hidden one is less” The change point is exactly at x = 1/2. All that remains is to calculate the total probability of success with that strategy. The favorable area to the left of 1/2 is worth 3/8, and the favorable area to the right of 1/2 is also worth 3/8. Therefore, the total probability is 3/8 + 3/8 = 3/4. So the optimal strategy is to compare the number seen with 1/2, and with it you get exactly 3/4 right.

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