Problem:
A pair of fair -sided dice is rolled times. What is the least value of such that the probability that the sum of the numbers face up on a roll equals at least once is greater than
Answer Choices:
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Solution:
There are equally likely outcomes for the numbers face up on the two dice. In 6 of those cases, the sum will be 7 , namely , and . Therefore the probability of getting a sum of 7 on any particular roll is , and the probability of not getting a sum of 7 is . Because the outcomes of successive rolls are independent, the probability of not getting a sum of 7 on any of rolls is . It remains to determine the least value of such that this value is less than or equal to .
Note that , and . But , so the requested value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions