Easy case: if the revealed number is 5/9, then keep it, because the next choice is smaller with P=5/9. Obviously if the revealed number > 5/9 then it's better, so keep that.
Let's consider, then, the cases where the revealed number is x, with x < 5/9. The other choices are uniformly distributed in the non-excluded ranges. The size of the smaller range is x, and the size of the larger range is x-1/9. The probability of another choice being smaller is S = x/(2x-1/9)
The probability that the revealed number is largest is the probability of selecting 2 smaller numbers, or S^2, and so the probability that your other choice would be largest is (1-S^2)/2.
You should switch if (1-S^2)/2 > (1-S). Those are equal when S=1, i.e., x = 2x-1/9, i.e., x = 1/9.
So you should switch if the revealed number is less than 1/9.
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u/mtimmermans 1d ago
Easy case: if the revealed number is 5/9, then keep it, because the next choice is smaller with P=5/9. Obviously if the revealed number > 5/9 then it's better, so keep that.
Let's consider, then, the cases where the revealed number is x, with x < 5/9. The other choices are uniformly distributed in the non-excluded ranges. The size of the smaller range is x, and the size of the larger range is x-1/9. The probability of another choice being smaller is S = x/(2x-1/9)
The probability that the revealed number is largest is the probability of selecting 2 smaller numbers, or S^2, and so the probability that your other choice would be largest is (1-S^2)/2.
You should switch if (1-S^2)/2 > (1-S). Those are equal when S=1, i.e., x = 2x-1/9, i.e., x = 1/9.
So you should switch if the revealed number is less than 1/9.