2008 iTest Problems/Problem 12
Problem
One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is higher than Tony's. "Let's play once more," says Tony. Let be the probability that the difference between the outcomes of the two dice is again exactly (regardless of which of the boys rolls higher), where a and b are relatively prime positive integers. Find .
Solution
The successful outcomes are easy to count by hand. Below is a table (each row represents Tony’s outcome and each column represents Mitchell’s outcome) where dots signify a successful result.
1 | 2 | 3 | 4 | 5 | 6 | |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 |
The number of successful outcomes is , and the total number of possibilities is , so the probability that the positive difference of the two dice rolls is three is . Thus, .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 11 |
Followed by: Problem 13 | |
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