2023 AMC 12A Problems/Problem 5
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 3 cases where the running total will equal 3; one roll; two rolls; or three rolls:
Case 1:
The chance of rolling a running total of 
in one roll is 
.
Case 2:
The chance of rolling a running total of in two rolls is 
since the dice rolls are a 2 and a 1 and vice versa.
Case 3:
The chance of rolling a running total of 3 in three rolls is 
since the dice values would have to be three ones.
Using the rule of sum, 
.
~walmartbrian ~andyluo
Solution 2 (Slightly different to Solution 1)
There are 3 cases where the running total will equal 3.
Case 1: Rolling a one three times
Case 2: Rolling a one then a two
Case 3: Rolling a three immediately
The probability of Case 1 is 
, the probability of Case 2 is (
, and the probability of Case 3 is
Using the rule of sums, adding every case gives the answer
~DRBStudent
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2023 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 4 |
Followed by Problem 6 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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