2023 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2023 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12A solutions, or check the answer key.

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Concepts:dice (probability)casework

Difficulty rating: 1270

Janet rolls a standard $6$ -sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3?$

$\dfrac{2}{9}$

$\dfrac{49}{216}$

$\dfrac{25}{108}$

$\dfrac{17}{72}$

$\dfrac{13}{54}$

Solution:

The running total is increasing, so it hits $3$ exactly when one of these disjoint openings occurs: a first roll of $3;$ rolls $1,2;$ rolls $2,1;$ or rolls $1,1,1.$

Their probabilities are $\dfrac16+\dfrac1{36}+\dfrac1{36}+\dfrac1{216} =\dfrac{36+6+6+1}{216}=\dfrac{49}{216}.$

Thus, the correct answer is B.

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