In a game show, there is a game in which you have to order the value of three prizes in order of least expensive to most expensive. You have to get all three right in order to win.

The only problem is you have your spouse do all the shopping, so you only know that the first prize is between 500 and 2000, the second prize is between 1000 and 2500, and the third prize is between 1500 and 3000.

Which order should you put them in so that you have the highest probability of winning, and what is the probability that you will win using this arrangement?

The first prize is equally likely to lie in each of the price ranges 500-999, 1000-1499, and 1500-2000. The second prize is equally likely to be in the price ranges 1000-1499, 1500-1999, and 2000-2500. The third prize will fall with equal probability in the price ranges 1500-1999, 2000-2499, and 2500-3000.

Here is the complete combination table, listing the 1st prize price range, the 2nd prize price range, the 3rd prize price range, and the resulting ascending price order of the prizes, is:

0500-0999 1000-1499 1500-1999 123
0500-0999 1000-1499 2000-2499 123
0500-0999 1000-1499 2500-3000 123
0500-0999 1500-1999 1500-1999 123 or 132
0500-0999 1500-1999 2000-2499 123
0500-0999 1500-1999 2500-3000 123
0500-0999 2000-2500 1500-1999 132
0500-0999 2000-2500 2000-2499 123 or 132
0500-0999 2000-2500 2500-3000 123
1000-1499 1000-1499 1500-1999 123 or 213
1000-1499 1000-1499 2000-2499 123 or 213
1000-1499 1000-1499 2500-3000 123 or 213
1000-1499 1500-1999 1500-1999 123 or 132
1000-1499 1500-1999 2000-2499 123
1000-1499 1500-1999 2500-3000 123
1000-1499 2000-2500 1500-1999 132
1000-1499 2000-2500 2000-2499 123 or 132
1000-1499 2000-2500 2500-3000 123
1500-2000 1000-1499 1500-1999 213 or 231
1500-2000 1000-1499 2000-2499 213
1500-2000 1000-1499 2500-3000 213
1500-2000 1500-1999 1500-1999 any order
1500-2000 1500-1999 2000-2499 123 or 213
1500-2000 1500-1999 2500-3000 123 or 213
1500-2000 2000-2500 1500-1999 132 or 312
1500-2000 2000-2500 2000-2499 123 or 132
1500-2000 2000-2500 2500-3000 123

123 occurs 21 times
132 occurs 9 times
213 occurs 9 times
231 occurs 2 times
312 occurs 2 times
321 occurs 1 time

If you order the prizes in 123 order, you will win 47% of the time (the highest probablility).
If you order the prizes in 132 order, you will win 21% of the time.
If you order the prizes in 213 order, you will win 21% of the time.
If you order the prizes in 231 order, you will win 4% of the time.
If you order the prizes in 312 order, you will win 4% of the time.
If you order the prizes in 321 order, you will win 3% of the time.
Edited on October 16, 2003, 8:25 am

Posted by Dan on 2003-10-16 08:24:27