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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?
  
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<math>\mathrm{(A)}\  
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<math>\mathrm{(A)}\ \frac 1{12}
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\qquad\mathrm{(B)}\  
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\qquad\mathrm{(B)}\ \frac 13
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\qquad\mathrm{(C)}\  
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\qquad\mathrm{(C)}\ \frac 12
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\qquad\mathrm{(D)}\  
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\qquad\mathrm{(D)}\ \frac 7{12}
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\qquad\mathrm{(E)}\ </math>
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\qquad\mathrm{(E)}\ \frac 23</math>
  
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__TOC__
 
 
== Solution ==
 
== Solution ==
 
=== Solution 1 ===
 
=== Solution 1 ===
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=== Solution 2 ===
 
=== Solution 2 ===
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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 they at least one does is <math>1 - \frac 12 = \frac 12 \Rightarrow \mathrm{(C)}</math>.
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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>; using [[complementary counting]], the probability that at least one does is <math>1 - \frac 12 = \frac 12 \Rightarrow \mathrm{(C)}</math>.
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=== Solution 3 (Alcumus) ===
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The product will be a multiple of 3 if and only if at least one of the two rolls is a 3 or a 6. The probability that Juan rolls 3 or 6 is <math>2/8 = 1/4</math>. The probability that Juan does not roll 3 or 6, but Amal does is <math>(3/4) (1/3) = 1/4</math>. Thus, the probability that the product of the rolls is a multiple of 3 is <cmath>\frac{1}{4} + \frac{1}{4} = \boxed{\frac{1}{2}}.</cmath>
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~aopsav (Credit to AoPS Alcumus)
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2002|ab=B|num-b=15|num-a=17}}
 
{{AMC12 box|year=2002|ab=B|num-b=15|num-a=17}}
  
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[[Category:Introductory Algebra Problems]]
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[[Category:Introductory Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 09:26, 5 November 2023

Problem

Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?

$\mathrm{(A)}\ \frac 1{12} \qquad\mathrm{(B)}\ \frac 13 \qquad\mathrm{(C)}\ \frac 12 \qquad\mathrm{(D)}\ \frac 7{12} \qquad\mathrm{(E)}\ \frac 23$

Solution

Solution 1

On both dice, only the faces with the numbers $3,6$ are divisible by $3$. Let $P(a) = \frac{2}{8} = \frac{1}{4}$ be the probability that Juan rolls a $3$ or a $6$, and $P(b) = \frac{2}{6} = \frac 13$ that Amal does. By the Principle of Inclusion-Exclusion,

\[P(a \cup b) = P(a) + P(b) - P(a \cap b) = \frac{1}{4} + \frac{1}{3} - \frac{1}{4} \cdot \frac{1}{3} = \frac{1}{2} \Rightarrow \mathrm{(C)}\]

Alternatively, the probability that Juan rolls a multiple of $3$ is $\frac{1}{4}$, and the probability that Juan does not roll a multiple of $3$ but Amal does is $\left(1 - \frac{1}{4}\right) \cdot \frac{1}{3} = \frac{1}{4}$. Thus the total probability is $\frac 14 + \frac 14 = \frac 12$.

Solution 2

The probability that neither Juan nor Amal rolls a multiple of $3$ is $\frac{6}{8} \cdot \frac{4}{6} = \frac{1}{2}$; using complementary counting, the probability that at least one does is $1 - \frac 12 = \frac 12 \Rightarrow \mathrm{(C)}$.

Solution 3 (Alcumus)

The product will be a multiple of 3 if and only if at least one of the two rolls is a 3 or a 6. The probability that Juan rolls 3 or 6 is $2/8 = 1/4$. The probability that Juan does not roll 3 or 6, but Amal does is $(3/4) (1/3) = 1/4$. Thus, the probability that the product of the rolls is a multiple of 3 is \[\frac{1}{4} + \frac{1}{4} = \boxed{\frac{1}{2}}.\] ~aopsav (Credit to AoPS Alcumus)

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

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