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Difference between revisions of "2021 Fall AMC 10B Problems/Problem 14"

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==Problem 14==
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Una rolls <math>6</math> standard <math>6</math>-sided dice simultaneously and calculates the product of the <math>6{ }</math> numbers obtained. What is the probability that the product is divisible by <math>4?</math>
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<math>\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}</math>
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==Solution==
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We will first find the probability that the product is not divisible by 4. We have 2 cases.
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Case 1: The product is not divisible by <math>2</math>.
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We need every number to be odd, and since the chance we roll an odd number is <math>\frac12,</math> our probability is <math>\left(\frac12\right)^6=\frac1{64}.</math>
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Case 2: The product is divisible by <math>2</math>, but not by <math>4</math>.
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We need <math>5</math> numbers to be odd, and one to be divisible by <math>2</math>, but not by <math>4</math>. There is a <math>\frac12</math> chance that an odd number is rolled, a <math>\frac13</math> chance that we roll a number satisfying the second condition (only <math>2</math> and <math>6</math> work), and 6 ways to choose the order in which the even number appears.
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Our probability is <math>\left(\frac12\right)^5\left(\frac13\right)\cdot6=\frac1{16}.</math>
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Therefore, the probability the product is not divisible by <math>4</math> is <math>\frac1{64}+\frac1{16}=\frac{5}{64}</math>.
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Our answer is <math>1-\frac{5}{64}=\boxed{\textbf{(E)}\frac{59}{64}}</math>.
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=15|num-b=13}}
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=15|num-b=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 08:26, 23 November 2021

Problem 14

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$

$\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Solution

We will first find the probability that the product is not divisible by 4. We have 2 cases.

Case 1: The product is not divisible by $2$.

We need every number to be odd, and since the chance we roll an odd number is $\frac12,$ our probability is $\left(\frac12\right)^6=\frac1{64}.$


Case 2: The product is divisible by $2$, but not by $4$.

We need $5$ numbers to be odd, and one to be divisible by $2$, but not by $4$. There is a $\frac12$ chance that an odd number is rolled, a $\frac13$ chance that we roll a number satisfying the second condition (only $2$ and $6$ work), and 6 ways to choose the order in which the even number appears.

Our probability is $\left(\frac12\right)^5\left(\frac13\right)\cdot6=\frac1{16}.$

Therefore, the probability the product is not divisible by $4$ is $\frac1{64}+\frac1{16}=\frac{5}{64}$.

Our answer is $1-\frac{5}{64}=\boxed{\textbf{(E)}\frac{59}{64}}$.

See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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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