2023 AMC 12A Problems/Problem 7

Problem

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 $3$ ?

Solution 1

There are $4$ cases where her running total can equal $3$ :

1. She rolled $1$ for three times consecutively from the beginning. Probability: $\frac{1}{6^3} = \frac{1}{216}$

2. She rolled a $1$ , then $2$ . Probability: $\frac{1}{6^2} = \frac{1}{36}$

3. She rolled a $2$ , then $1$ . Probability: $\frac{1}{6^2} = \frac{1}{36}$

4. She rolled a $3$ at the beginning. Probability: $\frac{1}{6}$

Add them together to get $\boxed{\textbf{(B)} \frac{49}{216}}.$

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2023 AMC 12A Problems/Problem 7

Problem

Solution 1