Difference between revisions of "2005 AMC 10B Problems/Problem 19"

 
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== Problem ==
 
== Problem ==
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One fair die has faces <math>1</math>, <math>1</math>, <math>2</math>, <math>2</math>, <math>3</math>, <math>3</math> and another has faces <math>4</math>, <math>4</math>, <math>5</math>, <math>5</math>, <math>6</math>, <math>6</math>. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?
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<math>\mathrm{(A)} \frac{1}{3} \qquad \mathrm{(B)} \frac{4}{9} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{5}{9} \qquad \mathrm{(E)} \frac{2}{3} </math>
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== Solution ==
 
== Solution ==
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For the sum to be odd, the resulting numbers must be of different parity. The probability that the first die is even and the second die is odd is <math>\frac{1}{3}*\frac{1}{3}=\frac{1}{9}</math> and the probability that the first die is odd and the second die is even is <math>\frac{2}{3}*\frac{2}{3}=\frac{4}{9}</math>. Therefore the probability that the dies have opposing parities (and consequently their sum is odd) is <math>\frac{1}{9}+\frac{4}{9}=\frac{5}{9}\Rightarrow \boxed{\mathrm{(D)}}</math>.
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== See Also ==
 
== See Also ==
 
*[[2005 AMC 10B Problems]]
 
*[[2005 AMC 10B Problems]]
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[[Category:Introductory Combinatorics Problems]]

Revision as of 14:15, 7 February 2009

Problem

One fair die has faces $1$, $1$, $2$, $2$, $3$, $3$ and another has faces $4$, $4$, $5$, $5$, $6$, $6$. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

$\mathrm{(A)} \frac{1}{3} \qquad \mathrm{(B)} \frac{4}{9} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{5}{9} \qquad \mathrm{(E)} \frac{2}{3}$

Solution

For the sum to be odd, the resulting numbers must be of different parity. The probability that the first die is even and the second die is odd is $\frac{1}{3}*\frac{1}{3}=\frac{1}{9}$ and the probability that the first die is odd and the second die is even is $\frac{2}{3}*\frac{2}{3}=\frac{4}{9}$. Therefore the probability that the dies have opposing parities (and consequently their sum is odd) is $\frac{1}{9}+\frac{4}{9}=\frac{5}{9}\Rightarrow \boxed{\mathrm{(D)}}$.

See Also

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