Craps is a 1-player dice game that is played as follows: Roll two 6-sided dice; their sum becomes your "initial" roll. If this initial roll is 2, 3, or 12, you lose. If the initial roll is 7 or 11, you win. Otherwise, keep rolling the dice until you reroll you initial number (and win) or until you roll a 7 (and lose).

You're betting that your adversary is going to lose his game of craps, which should be a favorable bet for you. But you receive an anonymous tip that he's secretly loaded one of the dice, so that it will always come up 5. This increases his chances of winning to 2/3.

Having learned of his evil deed, you're going to secretly load his other die so as to minimize his chance of winning. With what probability should you load each of the six faces? And how does that change his probability of winning?

(In reply to Great Problem by RandyOrton)

While this is not the solution to your problem.  Here is how your gambling friends would answer this question.  They would weight down the 2 to increase the probability of rolling a seven.  Here is why they would do this: Obviously you would be betting the "DONT'S" which means you would be giving yourself a huge disadvantage on the come-out roll.  However, one a point was established you would then re-assert your advantage by then placing an odds bet (i.e. lay the odds to the maximum of the table limit most casinos offer 20 times odds).

Heres how it would work out in real life.  These figures are estimates: placing a $10 don't bet would lose 2 out of ever three times causing a net loss of $20.  However, once the point becomes established every third roll the good gambler would then place a large odds bet (assuming 20x odds) to win $200 plus $10 every third point.  The lay odds would be as follows: point 6 or 8 lay $240 to win $200.  Point of 5 or 9 lay $300 to win $200 and 4 or 10 $400 to win $200. 

As you can see in this senerio, the the only one who would lose out is the casino.  The reson being both the adversary and you could both benefit from this senerio.  Your adversary would of course not be playing any odds bets.  Both players would in fact make money.

 

 

Posted by Bryon on 2005-04-23 00:13:31