2023 AMC 12A Problem 5

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?

Show solution

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

\begin{gathered} \dfrac16+\dfrac1{36}+\dfrac1{36}+\dfrac1{216}\\ {}=\dfrac{36+6+6+1}{216}\\ {}=\dfrac{49}{216}. \end{gathered}

Thus, the correct answer is B.