Question

Suppose three dice are rolled. Under the condition that no two show the same face, what is the probability that one of the faces is an ace?

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{1}{6}\)

Hint:

Alternatively, since all 3 faces are distinct, we are selecting 3 unique numbers out of 6.
The probability that the number 1 is included in any randomly chosen subset of 3 unique numbers out of 6 is simply:
\[ \frac{\binom{5}{2}}{\binom{6}{3}} = \frac{10}{20} = \frac{1}{2} \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem can be modeled as a conditional probability question.
We are given a condition: "no two dice show the same face".
This restricts our sample space only to those outcomes where all three dice show unique, distinct numbers.

Step 2: Detailed Explanation:
Let us find the size of the restricted sample space, denoted by \(N(S)\).
Since three dice are rolled and all three must show different faces:
- The first die can show any of the 6 faces.
- The second die must show a different face, so it has 5 possibilities.
- The third die must show a face different from the first two, leaving 4 possibilities.
Thus, the total number of restricted outcomes is:
\[ N(S) = 6 \cdot 5 \cdot 4 = 120 \]
Now, let us find the number of favorable outcomes, denoted by \(N(A)\), where one of the three faces is an "ace" (the number 1).
Since one of the faces must be 1, we need to choose the remaining two unique faces from the other 5 numbers \(\{2, 3, 4, 5, 6\}\).
The number of ways to select 2 numbers from 5 is:
\[ \binom{5}{2} = 10 \]
Once we have our set of 3 unique numbers (which includes the number 1), these 3 numbers can be distributed among the three rolled dice in:
\[ 3! = 6 \text{ ways} \]
Therefore, the number of favorable outcomes is:
\[ N(A) = \binom{5}{2} \cdot 3! = 10 \cdot 6 = 60 \]
Finally, the conditional probability is:
\[ P(\text{Ace} \mid \text{all faces distinct}) = \frac{N(A)}{N(S)} = \frac{60}{120} = \frac{1}{2} \]

Step 3: Final Answer:
The correct option is (B).