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2023 AMC 12A Problems/Problem 7: Difference between revisions

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There are <math>4</math> cases where her running total can equal <math>3</math>:
There are <math>4</math> cases where her running total can equal <math>3</math>:
1. She rolled <math>1</math> for three times consecutively from the beginning. Probability: <math>frac{1}{6^3} = frac{1}{216}
1. She rolled <math>1</math> for three times consecutively from the beginning. Probability: <math>frac{1}{6^3} = frac{1}{216}</math>
2.  She rolled a </math>1<math>, then </math>2<math>. Probability: </math>frac{1}{6^2} = frac{1}{36}
2.  She rolled a <math>1</math>, then <math>2</math>. Probability: <math>frac{1}{6^2} = frac{1}{36}</math>
3.  She rolled a <math>2</math>, then <math>1</math>. Probability: <math>frac{1}{6^2} = frac{1}{36}
3.  She rolled a <math>2</math>, then <math>1</math>. Probability: <math>frac{1}{6^2} = frac{1}{36}</math>
4.  She rolled a </math>3<math> at the beginning. Probability: </math>frac{1}{6}
4.  She rolled a <math>3</math> at the beginning. Probability: <math>frac{1}{6}</math>


Add them together to get <math>\boxed{textbf{(B)} frac{49}{216}}</math>
Add them together to get <math>\boxed{\textbf{(B)} \frac{49}{216}}.</math>
 
~d_code

Revision as of 19:57, 9 November 2023

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 $\boxed{\textbf{(B)} \frac{49}{216}}.$

~d_code

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