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Difference between revisions of "2012 AMC 10A Problems/Problem 9"

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== Solution ==
 
== Solution ==
  
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The total number of combinations when rolling two dice is <math>6*6 = 36</math>.
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The total number of combinations when rolling two dice is <math>6 \cdot 6 = 36</math>.
  
 
There are three ways that a sum of 7 can be rolled. <math>2+5</math>, <math>4+3</math>, and <math>6+1</math>. There are two 2's on one die and two 5's on the other, so there are a total of 4 ways to roll the combination of 2 and 5. There are two 4's on one die and two 3's on the other, so there are a total of 4 ways to roll the combination of 4 and 3. There are two 6's on one die and two 1's on the other, so there are a total of 4 ways to roll the combination of 6 and 1. Add <math>4 + 4 + 4 = 12</math>.
 
There are three ways that a sum of 7 can be rolled. <math>2+5</math>, <math>4+3</math>, and <math>6+1</math>. There are two 2's on one die and two 5's on the other, so there are a total of 4 ways to roll the combination of 2 and 5. There are two 4's on one die and two 3's on the other, so there are a total of 4 ways to roll the combination of 4 and 3. There are two 6's on one die and two 1's on the other, so there are a total of 4 ways to roll the combination of 6 and 1. Add <math>4 + 4 + 4 = 12</math>.

Revision as of 13:04, 8 September 2022

Problem

A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$

Solution

The total number of combinations when rolling two dice is $6 \cdot 6 = 36$.

There are three ways that a sum of 7 can be rolled. $2+5$, $4+3$, and $6+1$. There are two 2's on one die and two 5's on the other, so there are a total of 4 ways to roll the combination of 2 and 5. There are two 4's on one die and two 3's on the other, so there are a total of 4 ways to roll the combination of 4 and 3. There are two 6's on one die and two 1's on the other, so there are a total of 4 ways to roll the combination of 6 and 1. Add $4 + 4 + 4 = 12$.

Thus, our probability is $\frac{12}{36} = \frac{1}{3}$. The answer is $\boxed{\textbf{(D)}\ \frac13}$.

Solution 2

Assume we roll the die with only evens first. For whatever value rolled, there are exactly 2 faces on the odd die that makes the sum 7. The odd die has 6 faces, so our probability is $\boxed{\textbf{(D)}\ \frac13}$.

See Also

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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