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?

"Half" the time the 3 dice are different, really 120 of 216, or 56%.  Chance of winning here is 1/3.

"Half" the time the dice have two the same, really 90 of 216, or 42%.  Chance of winning here is 2/3.

Also, a small chance when all three dice are the same, really 6 of 216, or 3%.  Chance of winning here is 1.

120(1/3)+90(2/3)+6(1)=106 which is less than 108.

Or: 56%(1/3)+42%(2/3)+3%(1)=49.7%, less than 50%.

 

Posted by bernie on 2004-10-11 07:04:49