Question

Three fair dice are thrown simultaneously. What is the probability that the sum is 5?

• A. 1/36
• B. 1/6
• C. 1/54
• D. 1/72

Hint:

For probability problems with dice, first calculate the total possible outcomes, then count the favorable outcomes to find the probability.

Answer 1

The Correct Option is A

Step 1: Understanding the problem.
We are asked to find the probability that the sum of the numbers rolled on three fair dice is 5.
Step 2: Total possible outcomes.
Each die has 6 faces, so the total number of possible outcomes when three dice are thrown is \(6 \times 6 \times 6 = 216\).
Step 3: Finding favorable outcomes.
To get a sum of 5, the following combinations of dice rolls are possible: (1, 1, 3), (1, 2, 2), (2, 1, 2), (3, 1, 1). There are 6 different combinations that add up to 5.
Step 4: Probability calculation.
The probability is the number of favorable outcomes divided by the total number of outcomes: \[ P(\text{sum of 5}) = \frac{6}{216} = \frac{1}{36}. \] Step 5: Conclusion.
The correct answer is (A) 1/36, which is the probability that the sum of the dice is 5.

Answer 2

Since three dice give a small range of outcomes, another reliable method is to list every combination of three die values that adds up to 5, then count the ordered outcomes directly.

1. 1/36: Listing all unordered combinations of three numbers from 1-6 summing to 5 gives only two sets: (1,1,3) and (1,2,2). The set (1,1,3) can be ordered in 3 distinct ways among the three dice, and (1,2,2) can also be ordered in 3 distinct ways, giving \( 3+3=6 \) favorable outcomes out of \( 6^3=216 \) total outcomes, which simplifies to \( \frac{6}{216}=\frac{1}{36} \), matching this option exactly.
2. 1/6: This would require 36 favorable outcomes, far more than the 6 actually found by listing, so this overstates the probability.
3. 1/54: This corresponds to 4 favorable outcomes, which is fewer than the 6 combinations actually found, so it understates the count.
4. 1/72: This corresponds to 3 favorable outcomes, again fewer than the 6 valid combinations that sum to 5, so it also understates the true count.

Direct listing confirms exactly 6 favorable outcomes out of 216.

Therefore, the correct answer is 1/36.