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A Game of Probabilities


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18 replies to this topic

#11 bonanova

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Posted 23 June 2012 - 01:55 AM

Spoiler for my guess


Generalizing to j and k,
Spoiler for looks like

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#12 bushindo

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Posted 24 June 2012 - 02:32 AM

You and I are playing a little game (that is in no way whatsoever related to any game played on this site *cough*). There are 12 objects, 4 of which have the property M. The goal is to choose an object with the property M.

However, you and I are playing by different rules. For each round, first there is the N phase, in which you secretly choose an object (write down its number or something), then there is the D phase, during which I pick an object and it is revealed whether or not the object has property M.

If the object I pick has the property M, I win, if not we keep going. During any N phase, after you choose an object, you can call 'STOP' and the game will end. Then you reveal your choices, and if at least one has the property M, you win. If none have the property M, I win.

What is your play?

Bonus: Generalize to a case of n objects, k of which have the property M.


Here's an approximative strategy,

Spoiler for

Edited by bushindo, 24 June 2012 - 02:41 AM.

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#13 Yoruichi-san

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Posted 24 June 2012 - 03:28 AM

Excellent, thank you Bushindo :)

Spoiler for In agreement

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#14 dpalmer

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Posted 24 June 2012 - 04:37 AM

Spoiler for A solution and an extension

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#15 dpalmer

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Posted 24 June 2012 - 04:41 AM

Oh, I was assuming that the choice was only secret for D, but N knows what he or she draws. Those are two different games.
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#16 voider

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Posted 24 June 2012 - 05:01 AM

Consider the decision tree.
You can calculate the probability that you will win, when you predetermine on which round you will say STOP (if you survive until that round).
The first round probability is 4/12=0.33
Second is 0.397979...
This increases, until some round to say STOP where the probability of winning will begin to drop.
Obviously the "general solution" is to say STOP on the round that produces the overall highest probability of winning relative to the beginning of the decision tree.
My solution would be a computation, rather than a calculation. There are multiple ways of presenting it, but none of them would look nice.
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#17 voider

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Posted 24 June 2012 - 05:08 AM

The best strategy for N is to yell STOP only and as soon as he or she recieves something with property M...

You can't say STOP after they've already won.
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#18 dpalmer

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Posted 24 June 2012 - 05:17 AM

Spoiler for Another solution

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#19 dpalmer

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Posted 24 June 2012 - 05:21 AM

You can't say STOP after they've already won.

Of course not, but it doesn't matter. If N knows what he or she draws, then the only reason to continue the game is if he or she draws not-M. Calling STOP at this point would ensure a loss. Continuing gives a chance of victory and a chance of loss. So clearly it's better to allow another round. This logic continues until k=n, at which point whoever draws next wins.
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