A "friend" offers to play the following game: you throw a die, and he throws two dice. If both his dice are either higher or lower than yours, he wins; otherwise, you win.

First, you reason: out of three dice, one will always be the "middle" one, and only one out of three times it will be mine, so my odds are just 1/3 -- I shouldn't play.

After a while, you realize that you forgot about duplicate numbers. About 50% of the time, all three dice will be different, and then you have 1/3 chance of winning. But on the other 50%, you assuredly win, so the game stands 2/3 in your favor.

It's clear that BOTH lines of reasoning cannot be right, if any. Should you play, or shouldn't you?

Note: you can solve this mathemathically, or you can use "lateral thinking"; can you find both ways?

Math solution was easy:

Pr(me winning)

= Pr(me rolling 1 and winning)+Pr(me rolling 2 and winning)+...

= 1/6x11/36 + 1/6x19/36 + 1/6x23/36 + ...

= 106/216

= 49.07%

But my lateral brain has shut down!

Tried thinking of rolling 3 die - to win I have to split the other two numbers or equal the other two numbers. But I can never split if two rolls (my friend's) are the same - so there is a very small shift in the evenness of odds towards my friend.

This lateral theory is a little shakey...

Posted by David on 2004-07-08 21:35:16