Difference between revisions of "2002 AMC 12B Problems/Problem 16"
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Juan rolls a fair regular [[octahedron|octahedral]] die marked with the numbers <math>1</math> through <math>8</math>. Then Amal rolls a fair six-sided die. What is the [[probability]] that the product of the two rolls is a multiple of 3? | Juan rolls a fair regular [[octahedron|octahedral]] die marked with the numbers <math>1</math> through <math>8</math>. Then Amal rolls a fair six-sided die. What is the [[probability]] that the product of the two rolls is a multiple of 3? | ||
| − | <math>\mathrm{(A)}\ | + | <math>\mathrm{(A)}\ \frac 1{12} |
| − | \qquad\mathrm{(B)}\ | + | \qquad\mathrm{(B)}\ \frac 13 |
| − | \qquad\mathrm{(C)}\ | + | \qquad\mathrm{(C)}\ \frac 12 |
| − | \qquad\mathrm{(D)}\ | + | \qquad\mathrm{(D)}\ \frac 7{12} |
| − | \qquad\mathrm{(E)}\ </math> | + | \qquad\mathrm{(E)}\ \frac 23</math> |
__TOC__ | __TOC__ | ||
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=== Solution 2 === | === Solution 2 === | ||
| − | The probability that neither Juan nor Amal rolls a multiple of <math>3</math> is <math>\frac{6}{8} \cdot \frac{4}{6} = \frac{1}{2}</math>; by the [[complement principle]], the probability that | + | The probability that neither Juan nor Amal rolls a multiple of <math>3</math> is <math>\frac{6}{8} \cdot \frac{4}{6} = \frac{1}{2}</math>; by the [[complement principle]], the probability that at least one does is <math>1 - \frac 12 = \frac 12 \Rightarrow \mathrm{(C)}</math>. |
== See also == | == See also == | ||
Revision as of 17:38, 16 January 2008
Problem
Juan rolls a fair regular octahedral die marked with the numbers 
through 
. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?
Solution
Solution 1
On both dice, only the faces with the numbers 
are divisible by 
. Let 
be the probability that Juan rolls a or a 
, and 
that Amal does. By the Principle of Inclusion-Exclusion,
![\[P(a \cup b) = P(a) + P(b) - P(a \cap b) = \frac{1}{4} + \frac{1}{3} - \frac{1}{4} \cdot \frac{1}{3} = \frac{1}{2} \Rightarrow \mathrm{(C)}\]](http://latex.artofproblemsolving.com/c/c/2/cc29f83d1e2747949dccdb55b4eb258c113092a4.png)
Alternatively, the probability that Juan rolls a multiple of is 
, and the probability that Juan does not roll a multiple of but Amal does is 
. Thus the total probability is 
.
Solution 2
The probability that neither Juan nor Amal rolls a multiple of is 
; by the complement principle, the probability that at least one does is 
.
See also
| 2002 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 15 |
Followed by Problem 17 |
| 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 12 Problems and Solutions | |
