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General Gambling >> Probability
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RocketManJames
member

Reged: 11/06/02
Posts: 118
09/30/04 02:26 PM

A friend brought this up, and I am pretty sure I know why this doesn't work, but I wanted to throw this out at all of you.

Two distinct real numbers are chosen randomly from the universe of real numbers. So we have number A and number B.

Now, your goal is to pick the bigger number. You are allowed to look at A... then you must decide if you want to keep it or take number B instead.

This is obviously a 50/50 proposition.

Now, here's a strategy that seems to give you better than 50/50. It shouldn't work, but it's sort of clever.

Choose any number... let's call it C.

Now, when you look at A, you compare it to C. If A is bigger than C, take A, otherwise take B.

There is some probability P_bigger that your number C is bigger than A and B. There is some P_smaller that your number C is smaller than both A and B. There is some P_between that C is between A and B.

So, using this strategy, the probability that you pick the bigger number is:

P_smaller * 0.5 + P_bigger * 0.5 + P_between * 1.00

which is

P_between + (1 - P_between) * 0.5

So, only if P_between is 0, do we end up with 50/50. If P_between is anything greater than 0, then we have better than 50% chance at picking the bigger number.

In Edit: In case it wasn't clear... the reason why P_between is 100% probability, is that if your number C is between A and B, then you are guaranteed to choose the bigger number following the above strategy as A relates to C.

This can't be right, because you cannot create information out of non-information.

I am pretty sure why this doesn't work has to do with the fact that the random numbers A and B are chosen from all real numbers. If it were chosen from a bounded set of numbers then the strategy does give you a boost in probability.

Any thoughts?

-RMJ

Edited by RocketManJames (09/30/04 02:40 PM)

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