Difference between revisions of "1989 AIME Problems/Problem 5"
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Latest revision as of 00:07, 25 June 2023
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
. To find the probability getting
heads, we need to find
(recall that
cannot be
). 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 |
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