(AHSME 1996) When n standard 6 sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same probability of obtaining a sum of S. What is the smallest possible value of S?
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Observe that for any natural number a:
For 2a dice (even):
prob. of obtaining a sum of 12a (maximum sum) = prob. of obtaining a sum of 2a (minimum sum)
prob. of obtaining a sum of 12a -1 = prob. of obtaining a sum of 2a+1
prob. of obtaining a sum of 12a -2 = prob. of obtaining a sum of 2a+2
.
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prob. of obtaining a sum of 12a-(5a-1) = prob. of obtaining a sum of 2a+(5a-1)
prob. of obtaining a sum of 12a-(5a) = prob. of obtaining a sum of 2a+(5a)
For 2a+1 dice (odd):
prob. of obtaining a sum of 12a+6 (maximum sum) = prob. of obtaining a sum of 2a+1 (minimum sum)
prob. of obtaining a sum of (12a+6)-1 = prob. of obtaining a sum of (2a+1)+1
prob. of obtaining a sum of (12a+6)-2 = prob. of obtaining a sum of (2a+1)+2
.
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prob. of obtaining a sum of (12a+6)-(5a+1) = prob. of obtaining a sum of (2a+1)+(5a+1)
prob. of obtaining a sum of (12a+6)-(5a+2) = prob. of obtaining a sum of (2a+1)+(5a+2)
Based on these observations,
*In order to minimize S, 1994 should be one of the largest (or maximum) sum/s
*Note that the maximum sums, 12a and 12a+6, are both divisible by 6 for any natural number a. Since 1994 is not divisible by 6, it is not the maximum sum
*The nearest natural number which is divisible by 6 and greater than 1994 is 1998. Hence 1998 is the maximum sum. So 12a = 1998 (if even) or 12a+6 =1998 (if odd).
12a = 1998…a =166.5 (not a natural number), 12a+6 = 1998… a = 166 (natural number)
*So there are odd numbers of dice. Since 1998-1994 = 4, then 1994 should be (12a+6)-4. And since prob. of obtaining a sum of (12a+6)-4 = prob. of obtaining a sum of (2a+1) +4, then prob. of obtaining a sum of 1994 is the same prob. of obtaining a sum of [2(166) +1] +4 = 337
Quote:
“There is no more difficult art to acquire than the art of observation, and for some men it is quite as difficult to record an observation in brief and plain language.” – William Osler
