Sneaky Joe has just invited you as a VIP to his new casino. You know this is probably an attempt steal your money, for he always find ways to swindle people. However, you go anyway.

When you get there, he says, "Come over here and join me in a game of craps." You become slightly suspicious, but agree to come anyway. When you go over, he says, "OK, here's how we play craps in this casino, 'cause it's different here than other casinos. You have 3 dice, 2 of them are 12-sided dice and another is a 40-sided die. I will roll the 2 12-sided dice. Then you roll the 40-sided one. If your number is between (y^2-x) and (x^2-y) inclusively, being that x=the number I got from the first roll and y=the number I got on the second, you will win $10. Otherwise, you will lose $10."

"Ok," you think, "I'm pretty sure that the odds are against me, especially if it's a game that Joe made himself. But I need $30, and I only have $10." So, what's the probability of you winning $30 (as in $30 in the black, without any debt, which included the original $10 paid) from this game?

(NOTE: It can be done WITHOUT trial and error, and it is my request, though you do not have do it, that you solve this without trial and error.)

(In reply to re(4): Solution by Penny)

"So it depends on how the word "between" is interpreted."

Yes it does depend on the interpretation of this word.  The clock example of "9 to 5" of course adds the complication of a cyclic (modular) arithmetic.  The numbers here are not modular, so according to one interpretation (the stricter), "between 100 and 25 inclusive" would actually include no numbers at all, as a candidate number must be >= 100 and at the same time <= 25, which no number can satisfy.  Likewise "between 10th Avenue and 7th Avenue" would not include 9th Avenue (or any avenue), except if interpreted geographically rather than numerically (in the same way as "between Maple Street and Elm Street").

If Joe tried that on me, I'd be pretty mad at him and his interpretation.  No wonder Victor is suspicious of this sneaky character.

Posted by Charlie on 2004-04-18 10:04:57