In a game show, there is a certain game in which there are four hidden digits. There are no numbers greater than six among them, and no zeros.

You roll a die and then guess if the first digit is higher or lower than what you rolled. (If the die you rolled is equal to the first digit, you win no matter what you said.) You then roll and guess for each of the other three digits.

If you use the best strategy each time when saying "higher" or "lower", what is the chance you will get all four right and win? (Keep in mind you have no idea what the 4 digit number is.)

The strategy is "if you roll 1, 2, or 3, call 'higher'; otherwise, call 'lower'.".

If you roll a 1 or 6, you win.
If you roll a 2 or 5, you lose only to a 1 or 6.
If you roll a 3 or 4, you lose to 1 and 2, or 5 and 6.

The chances of losing are (0+0+1+1+2+2)/36= 1/6. Thus, your chances of winning are 5/6 for each digit, or (5/6)^4 for the 4 digits.

Posted by Old Original Oskar! on 2005-11-04 08:27:19