1989 AIME Problems/Problem 5
Problem
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to and is the same as that of getting heads exactly twice. Let , in lowest terms, be the probability that the coin comes up heads in exactly out of flips. Find .
Solution
Solution 1
Denote the probability of getting a heads in one flip of the biased coin as . Based upon the problem, note that . After canceling out terms, we get , so . The answer we are looking for is , so .
Solution 2
Denote the probability of getting a heads in one flip of the biased coins as and the probability of getting a tails as . Based upon the problem, note that . After cancelling out terms, we end up with (recall that cannot be 0). To find the probability getting 3 heads, we need to find . The result after simplifying is , so .
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.