 ## The Guessing Games

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Edited by Richard Walker, Saturday, 18 Sep 2021, 00:27

A friend tells you she has chosen a number from 1 to 100, and challenges you to guess what it is. To help you a little, she says you can ask exactly one question about the number and she will answer it honestly. Armed with the extra information her answer provides, you can then proceed to guess what number she picked.

[Adapted from a puzzle by Alex Besos.]

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Hi Richard,

This question won't let me go so I'll venture the answer/question below. I suspect it's not the one you have in mind but however!!!

Is it the largest prime number within that range?

Joseph.

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That will do but there are other answers equally good.

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Consider your question: what is the probability they answer yes (1/100) and what is the probability you will guess correctly in that case? That must be 1/1 given your question.

But what is the probability they answer no, and what is the probability you will guess correctly given that information?

So overall what is the probability of success? I find it sort of obvious but not at all intuitive, so not obvious either.

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Hi Richard,

A lot of thought-provoking, engaging and deceptively deeper than what might at first, to me, be thought/expected in your answer. Really interesting.

Many thanks,

Joseph.

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Yes, so with your example we have

They say yes with probability 1/100 and then it is 1/1 you will guess the number so (1/100) x (1/1) = 1/100

They say no with probability 99/100 and then it is 1/99 you will guess the number so (99/100) x (1/99) = 1/100

Since the two cases are independent we can add the probabilities 1/100 + 1/100 = 1/50.

But in fact any question will come to the same result, 1/50 (just so long as the answer is not a foregone conclusion, which would be the case with questions such as "Is it 200" to which the answer is bound to be "No", or "Is it between 1 and 100 inclusive", to which the answer is bound to be "Yes". Questions like these yield no new information so don't narrow the field.)

But any other question will give 1/50. For example "Is it divisible by 10?".

They say yes with probability 10/100 and then it is 1/10 you will guess the number so (10/100) x (1/10) = 1/100

They say no with probability 90/100 and then it is 1/90 you will guess the number so (90/100) x (1/90) = 1/100

1/100 + 1/100 = 1/50 as before.

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Hi Richard,

As I expected, the answer to your original posting is a lot more complex than what might be thought at first reading - indeed even after further readings. I nevertheless follow your explanations and appreciate the stimulation they give.

Regards,

Joseph.