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^{}_{}</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>. | + | 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 == | ||
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== 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]] | ||
Revision as of 18:51, 23 October 2007
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
Denote the probability of getting a heads in one flip of the biased coins as 
. Based upon the problem, note that 
. After canceling out terms, we get 
, so 
. The answer we are looking for 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 | ||
