Carnie Val ran a game of chance on the boardwalk at the Jersey Shore. I won't mention what beach. The player would plunk down his dollar and an array of 15 lights arranged in an equilateral triangle would start to flash. After a few seconds, three chosen at random would remain lit. If the three lit bulbs formed the vertices of an equilateral triangle, the lucky player would win a fuzzy stuffed animal. The game was on the up-and-up in the sense that any combination of lights was as likely to turn up as any other. For convenience of discussion, the bulbs are numbered as follows:

               1
               
             2   3
           
           4   5   6
          
         7   8   9  10
         
       11  12  13  14 15

One day, one of the lights failed to work. It was taken out of the random cycle, so that at the end three of the remaining 14 lights would stay lit, again with equal likelihood of any of the possible arrangements.

Val has no incentive to fix the broken light, as the new arrangement gives the player a lower probability of winning. That probability is the reciprocal of an integer, that is 1 over a whole number.

What is that probability?

If I did my math correctly, there are 27 possible winning combinations when all 15 lights are working. With only 14 lights working, the total number of possible combinations is 364 (14_C_3).

New number of winning combinations with 14 working lights would depend on which light is out.

If 1, 11, or 15 were out, only 4 winning combinations would be removed. Winning probability is 23/364.

If 2, 3, 7, 10, 12, or 14 were out, only 5 winning combinations would be reduced. Winning probability is 22/364 or 11/182.

If 4, 6, or 13 were out, only 6 winning combinations would be reduced. Winning probablility is 21/364 or 3/52.

If 5, 8, or 9 were out, 13 possible winning combinations would be removed. Winning probability is 13/364 or 1/26.

Posted by Michael on 2008-02-04 15:54:22