Difference between revisions of "2014 AMC 10B Problems/Problem 16"

Revision as of 17:13, 20 August 2021

Problem

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

$\textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9} \qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}$

Solution

We split this problem into $2$ cases.

First, we calculate the probability that all four are the same. After the first dice, all the numbers must be equal to that roll, giving a probability of $1 \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{216}$ .

Second, we calculate the probability that three are the same and one is different. After the first dice, the next two must be equal and the third different. There are $4$ orders to roll the different dice, giving $4 \cdot 1 \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} \cdot \dfrac{5}{6} = \dfrac{5}{54}$ .

Adding these up, we get $\dfrac{7}{72}$ , or $\boxed{\textbf{(B)}}$ .

Solution 2

Note that there are two cases for this problem

$\textbf{Case 1}$ : Exactly three of the dices show the same value.

There are $5$ values that the remaining die can take on, and there are $\binom{4}{3}=4$ ways to choose the die. There are $6$ ways that this can happen. Hence, $6\cdot 4\cdot5=120$ ways.

$\textbf{Case 2}$ : Exactly four of the dices show the same value.

This can happen in $6$ ways.

Hence, the probability is $\frac{120+6}{6^{4}}=\frac{21}{216}\implies \frac{7}{72}\implies \boxed{\textbf{(B)}}$

Solution 3

We solve using PIE.

We first calculate the number of ways that we can have $3$ dice be the same and the other dice be anything. We therefore have $\binom{4}{3} \cdot 6 \cdot 6 = 144$ ways to have at least $3$ dice be the same.

But wait! We have overcounted the case where all $4$ dice are the same! Since the previous case occurs in each of these cases $4$ times, we must subtract the $4$ -dice total three times in order to have them counted once. There are $6$ ways to have four dice be the same, so we our total count is $144 - 3(6) = 126$ .

Therefore, our probability is $\frac{126}{6^4} = \boxed{\frac{7}{72}}$ , which is answer choice $\boxed{\textbf{(B)}}$ .

-FIREDRAGONMATH16

@@ Line 8: / Line 8: @@
 We split this problem into <math>2</math> cases.
-First, we calculate the probability that all four are the same. After the first dice, all the number must be equal to that roll, giving a probability of <math>1 \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{216}</math>.
+First, we calculate the probability that all four are the same. After the first dice, all the numbers must be equal to that roll, giving a probability of <math>1 \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{216}</math>.
 Second, we calculate the probability that three are the same and one is different. After the first dice, the next two must be equal and the third different. There are <math>4</math> orders to roll the different dice, giving <math>4 \cdot 1 \cdot \dfrac{1}{6} \cdot \dfrac{1}{6} \cdot \dfrac{5}{6} = \dfrac{5}{54}</math>.

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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