Difference between revisions of "2023 AMC 12A Problems/Problem 5"

Revision as of 02:57, 10 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?

$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

Solution

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 $3$ in one roll is $\frac{1}{6}$ .

Case 2: The chance of rolling a running total of $3$ in two rolls is $\frac{1}{6}\cdot\frac{1}{6}\cdot2=\frac{1}{18}$ 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 $\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{216}$ since the dice values would have to be three ones.

Using the rule of sum, $\frac{1}{6}+\frac{1}{18}+\frac{1}{216}=\boxed{\textbf{(B) }\frac{49}{216}}$ .

~walmartbrian ~andyluo ~DRBStudent

Solution 2 (Brute Force)

Because there is only a maximum of 3 rolls we must count (running total = 3 means there can't be a fourth roll counted), we can simply list out all of the probabilities.

If we roll a 1 on the first, the rolls that follow must be 2 or {1,1}, with the following results not mattering. This leaves a probability of $\frac{1}{6}\times\frac{1}{6} + \frac{1}{6}\times\frac{1}{6}\times\frac{1}{6} = \frac{7}{216}$ .

If we roll a 2 on the first, the roll that follows must be 2, resulting in a probability of $\frac{1}{6}\times\frac{1}{6} = \frac{1}{36}$ .

If we roll a 3 on the first, the rolls that follow do not matter, resulting in a probability of $\frac{1}{6}$ . Any roll greater than three will result in a running total greater than 3 no matter what, so those cases can be ignored. Summing the answers, we have $\frac{7}{36} + \frac{1}{216} + \frac{1}{36} = \frac{42+1+6}{216} = \boxed{\textbf{(B) }\frac{49}{216}}$ .

~Failure.net

Solution 3 (same as ~walmartbrian ~andyluo ~DRBStudent's first solution with slightly different explanation)

Can we get a 3 in 1 roll? Sure but only if we get a 3, which has probability 1/6
Can we get a 3 in 2 rolls? There are 36 (6x6) possible rolls, 2 of which (1 and 2, 2 and 1) make 3, so probability is 2/36 = 1/18
Can we get a 3 in 3 rolls? There's only 1 way - to get 3 1s, so probability is (1/6)**3
Can we get a 3 in 4 rolls? No - the minimum roll would be 1+1+1+1 = 4

So our answer is 1/6 (1 + 1/3 + 1/36) = 1/216 (36+12+1) = $\boxed{\textbf{(B) 49/216}}$

~Dilip

Solution 4

Consider sequences of 4 integers with each integer between 1 and 6, the number of permutations of 6 numbers is $6^4=1296$ .

The following 4 types of sequences that might generate a running total of the numbers to be equal to 3 (x, y, or z is any integer between 1 and 6).

Sequence #1, (1, 1, 1, x): there are 6 possible sequences.

Sequence #2, (1, 2, x, y): there are $6^2 = 36$ possible sequences.

Sequence #3, (2, 1, x, y): there are $6^2 = 36$ possible sequences.

Sequence #4, (3, x, y, z): there are $6^3 = 216$ possible sequences.

Out of 1296 possible sequences, there are a total of $6 + 36 + 36 + 216 = 294$ sequences that qualify. Hence, the probability is $294 / 1296 = \boxed{\textbf{(B) }\frac{49}{216}}$

~sqroot

@@ Line 49: / Line 49: @@
 ==Solution 4==
-Consider sequences of 4 numbers with each number between 1 and 6, the number of permutations of 6 numbers is <math>6^4=1296</math>.
+Consider sequences of 4 integers with each integer between 1 and 6, the number of permutations of 6 numbers is <math>6^4=1296</math>.
-There are 4 types of sequences that might generate a running total of the numbers to be equal to 3.
+The following 4 types of sequences that might generate a running total of the numbers to be equal to 3 (x, y, or z is any integer between 1 and 6).
-Sequence #1, (1, 1, 1, x) where x is any of the 6 numbers: there are 6 possible sequences.
+Sequence #1, (1, 1, 1, x): there are 6 possible sequences.
-Sequence #2, (1, 2, x, y) where x and y are any of the 6 numbers: there are <math>6^2 = 36</math> possible sequences.
+Sequence #2, (1, 2, x, y): there are <math>6^2 = 36</math> possible sequences.
-Sequence #3, (2, 1, x, y) where x and y are any of the 6 numbers: there are <math>6^2 = 36</math> possible sequences.
+Sequence #3, (2, 1, x, y): there are <math>6^2 = 36</math> possible sequences.
-Sequence #4, (3, x, y, z) where x, y and z are any of the 6 numbers: there are <math>6^3 = 216</math> possible sequences.
+Sequence #4, (3, x, y, z): there are <math>6^3 = 216</math> possible sequences.
-Therefore, out of 1296 possible sequences, there are a total of <math>6 + 36 + 36 + 216 = 294</math> sequences that qualify. Hence, the probability is <math>294 / 1296 = \boxed{\textbf{(B) }\frac{49}{216}}</math>
+Out of 1296 possible sequences, there are a total of <math>6 + 36 + 36 + 216 = 294</math> sequences that qualify. Hence, the probability is <math>294 / 1296 = \boxed{\textbf{(B) }\frac{49}{216}}</math>
 ~sqroot

2023 AMC 10A (Problems • Answer Key • Resources)
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2023 AMC 12A (Problems • Answer Key • Resources)
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