Difference between revisions of "1989 AIME Problems/Problem 5"
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=== Solution 2 === | === Solution 2 === | ||
− | Denote the probability of getting a heads in one flip of the biased coins as <math>h</math> and the probability of getting a tails as <math>t</math>. Based upon the problem, note that <math>{5\choose1}(h)^1(t)^4 = {5\choose2}(h)^2(t)^3</math>. After cancelling out terms, we end up with <math>t = 2h</math>. To find the probability getting 3 heads, we need to find <math>{5\choose3}\frac{(h)^3(t)^2}{(h + t)^5} =10\cdot\frac{(h)^3(2h)^2}{(h + 2h)^5}</math>. The result after simplifying is <math>\frac{40}{243}</math>, so <math>i + j = 40 + 243 = \boxed{283}</math>. | + | Denote the probability of getting a heads in one flip of the biased coins as <math>h</math> and the probability of getting a tails as <math>t</math>. Based upon the problem, note that <math>{5\choose1}(h)^1(t)^4 = {5\choose2}(h)^2(t)^3</math>. After cancelling out terms, we end up with <math>t = 2h</math> (recall that <math>h</math> cannot be 0). To find the probability getting 3 heads, we need to find <math>{5\choose3}\frac{(h)^3(t)^2}{(h + t)^5} =10\cdot\frac{(h)^3(2h)^2}{(h + 2h)^5}</math>. The result after simplifying is <math>\frac{40}{243}</math>, so <math>i + j = 40 + 243 = \boxed{283}</math>. |
== See also == | == See also == |
Revision as of 12:46, 23 April 2017
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 |
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