2021 AIME I Problems/Problem 1

Problem

Zou and Chou are practicing their 100-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

Solution 1(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.

The probability that Zou loses a race is $\frac13,$ and the probability that he wins the following race is $\frac13.$ For each of the three other races, the probability that Zou wins is $\frac23.$

There are four such outcome sequences. The probability of one such sequence is $\left(\frac13\right)^2\left(\frac23\right)^3.$

Case (2): Zou loses the last race.

The probability that Zou loses a race is $\frac13.$ For each of the four other races, the probability that Zou wins is $\frac23.$

There is one such outcome sequence. The probability is $\left(\frac13\right)^1\left(\frac23\right)^4.$

Answer

The requested probability is $4\left(\frac13\right)^2\left(\frac23\right)^3+\left(\frac13\right)^1\left(\frac23\right)^4=\frac{32}{243}+\frac{16}{243}=\frac{48}{243}=\frac{16}{81},$ and the answer is $16+81=\boxed{097}.$

~MRENTHUSIASM

Solution 2 (Casework but Bashier)

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=H17E9n2nIyY

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2021 AIME I Problems/Problem 1

Contents

Problem

Solution 1(Casework)

Solution 2 (Casework but Bashier)

Video Solution by Punxsutawney Phil

See also