Difference between revisions of "1989 AIME Problems/Problem 5"
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== Problem == | == Problem == | ||
− | When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to <math>0 | + | When a certain biased coin is flipped five times, the [[probability]] of getting heads exactly once is not equal to <math>0</math> and is the same as that of getting heads exactly twice. Let <math>\frac ij</math>, in lowest terms, be the probability that the coin comes up heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j</math>. |
== Solution == | == Solution == | ||
− | Denote the probability of getting a heads in one flip of the biased | + | ===Solution 1 === |
+ | Denote the probability of getting a heads in one flip of the biased coin as <math>h</math>. Based upon the problem, note that <math>{5\choose1}(h)^1(1-h)^4 = {5\choose2}(h)^2(1-h)^3</math>. After canceling out terms, we get <math>1 - h = 2h</math>, so <math>h = \frac{1}{3}</math>. The answer we are looking for is <math>{5\choose3}(h)^3(1-h)^2 = 10\left(\frac{1}{3}\right)^3\left(\frac{2}{3}\right)^2 = \frac{40}{243}</math>, so <math>i+j=40+243=\boxed{283}</math>. | ||
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+ | === 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 <math>3</math> heads, we need to find <math>{5\choose3}\dfrac{(h)^3(t)^2}{(h + t)^5} =10\cdot\dfrac{(h)^3(2h)^2}{(h + 2h)^5}</math> (recall that <math>h</math> cannot be <math>0</math>). The result after simplifying is <math>\frac{40}{243}</math>, so <math>i + j = 40 + 243 = \boxed{283}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1989|num-b=4|num-a=6}} | {{AIME box|year=1989|num-b=4|num-a=6}} | ||
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+ | [[Category:Intermediate Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:46, 27 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 . 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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