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> |
− | |||
== Solution == | == Solution == | ||
=== Solution 1 === | === Solution 1 === | ||
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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>; | + | 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>. |
+ | |||
+ | === 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 <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> | ||
+ | ~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}} | ||
− | [[Category:Introductory | + | [[Category:Introductory Combinatorics Problems]] |
+ | {{MAA Notice}} |
Latest revision as of 10:26, 5 November 2023
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,
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
; using complementary counting, the probability that at least one does is
.
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 . The probability that Juan does not roll 3 or 6, but Amal does is
. Thus, the probability that the product of the rolls is a multiple of 3 is
~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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