Difference between revisions of "2012 AMC 12A Problems/Problem 11"

Revision as of 09:11, 8 June 2024

Problem

Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$ , and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

$\textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1$

Solution 1

If $m$ is the probability Mel wins and $c$ is the probability Chelsea wins, $m=2c$ and $m+c=\frac12$ . From this we get $m=\frac13$ and $c=\frac16$ . For Alex to win three, Mel to win two, and Chelsea to win one, in that order, is $\frac{1}{2^3\cdot3^2\cdot6}=\frac{1}{432}$ . Multiply this by the number of permutations (orders they can win) which is $\frac{6!}{3!2!1!}=60.$

$\frac{1}{432}\cdot60=\frac{60}{432}=\boxed{\textbf{(B)}\ \frac{5}{36}}$

Solution 2

The probability that Alex wins is $\frac{1}{2}$ , the probability that Mel wins is $\frac{1}{3}$ , and the probability that Chelsea wins is $\frac{1}{6}$ . There are $\binom{6}{3}$ ways to pick the three games that Alex wins, and $\binom{3}{2}$ ways to pick 2 out of the 3 remaining games that Mel wins - theres $\binom{1}{1}$ ways to see that Chelsea wins. Therefore, the number of ways is

$\frac{1}{2^3}\times\frac{1}{3^2}\times\frac{1}{6}\times\binom{6}{3}\times\binom{3}{2} = \boxed{\textbf{(B)}\ \frac{5}{36}}$

~ youtube.com/indianmathguy

Video Solution by TheBeautyofMath

I preview the concept that will be used in the first part of this video, and show why it's important "Don't Memorize, Understand". I adapt the solution method of an older problem to inform the solution of this one: https://youtu.be/PO3XZaSchJc

~IceMatrix

@@ Line 5: / Line 5: @@
 <math> \textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1 </math>
-== Solution ==
+== Solution 1 ==
 If <math>m</math> is the probability Mel wins and <math>c</math> is the probability Chelsea wins, <math>m=2c</math> and <math>m+c=\frac12</math>. From this we get <math>m=\frac13</math> and <math>c=\frac16</math>. For Alex to win three, Mel to win two, and Chelsea to win one, in that order, is <math>\frac{1}{2^3\cdot3^2\cdot6}=\frac{1}{432}</math>. Multiply this by the number of permutations (orders they can win) which is <math>\frac{6!}{3!2!1!}=60.</math>
 <cmath>\frac{1}{432}\cdot60=\frac{60}{432}=\boxed{\textbf{(B)}\ \frac{5}{36}}</cmath>
+== Solution 2 ==
+The probability that Alex wins is <math>\frac{1}{2}</math>, the probability that Mel wins is <math>\frac{1}{3}</math>, and the probability that Chelsea wins is <math>\frac{1}{6}</math>. There are <math>\binom{6}{3}</math> ways to pick the three games that Alex wins, and <math>\binom{3}{2}</math> ways to pick 2 out of the 3 remaining games that Mel wins - theres <math>\binom{1}{1}</math> ways to see that Chelsea wins. Therefore, the number of ways is
+<cmath>\frac{1}{2^3}\times\frac{1}{3^2}\times\frac{1}{6}\times\binom{6}{3}\times\binom{3}{2} = \boxed{\textbf{(B)}\ \frac{5}{36}}</cmath>
+~ youtube.com/indianmathguy
 ==Video Solution by TheBeautyofMath==

2012 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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