2021 AIME I Problems/Problem 1
Problem
Zou and Chou are practicing their 100-meter sprints by running 
races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is 
if they won the previous race but only 
if they lost the previous race. The probability that Zou will win exactly 
of the races is 
, where 
and 
are relatively prime positive integers. What is 
?
Solution (Casework)
For the last five races, Zou wins four and loses one. There are five possible outcome sequences, and we will proceed by casework:
Case (1): Zou does not lose the last race.
There are four such outcome sequences. The probability of one such sequence is 
Case (2): Zou loses the last race.
There is one such outcome sequence. The probability is 
Answer
The requested probability is ![\[4\left(\frac23\right)^3\left(\frac13\right)^2+\left(\frac23\right)^4\left(\frac13\right)^1=\frac{32}{243}+\frac{16}{243}=\frac{48}{243}=\frac{16}{81},\]](http://latex.artofproblemsolving.com/7/d/7/7d7d0609b19157f6819398fd8fc1e30eb63a32c2.png)
and the answer is 
See also
| 2021 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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