Difference between revisions of "2021 AIME I Problems/Problem 1"
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There is only one possibility for this, so our probability is <math>\frac{16}{243}</math>. | There is only one possibility for this, so our probability is <math>\frac{16}{243}</math>. | ||
− | <b> Case 3: Neither happens <b> | + | <b> Case 3: Neither happens <\b> |
There are three ways that this happens. Each has one loss that is not the last race. Therefore, the probability that one happens is <math>\frac{1}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{8}{243}</math>. Thus, the total probability is <math>\frac{8}{243} \cdot 3 = \frac{24}{243}</math>. | There are three ways that this happens. Each has one loss that is not the last race. Therefore, the probability that one happens is <math>\frac{1}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{8}{243}</math>. Thus, the total probability is <math>\frac{8}{243} \cdot 3 = \frac{24}{243}</math>. |
Revision as of 11:20, 12 March 2021
Contents
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 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 and the probability that he wins the following race is
For each of the three other races, the probability that Zou wins is
There are four such outcome sequences. The probability of one such sequence is
Case (2): Zou loses the last race.
The probability that Zou loses a race is For each of the four other races, the probability that Zou wins is
There is one such outcome sequence. The probability is
Answer
The requested probability is and the answer is
~MRENTHUSIASM
Solution 2 (Casework but Bashier)
We have cases, depending on which race Zou lost. Let
denote a won race, and
denote a lost race for Zou. The possible cases are
. The first case has probability
. The second case has probability
. The third has probability
. The fourth has probability
. Lastly, the fifth has probability
. Adding these up, the total probability is
, so
. ~rocketsri
This is for if you're paranoid like me, and like to sometimes write out all of the cases if there are only a few.
Solution 3 (Even more Casework)
Case 1: Zou loses the first race <\b>
In this case, Zou must win the rest of the races. Thus, our probability is .
Case 2: Zou loses the last race <\b>
There is only one possibility for this, so our probability is .
Case 3: Neither happens <\b>
There are three ways that this happens. Each has one loss that is not the last race. Therefore, the probability that one happens is . Thus, the total probability is
.
Adding these up, we get .
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=H17E9n2nIyY
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.