Six fair dice are rolled . The probability that the product of the numbers appearing on top faces is prime is

Options

A. `1/2(1/6)^4`
B. `(1/2)^6`
C. `1/(6^4)`
D. `1/2(1/6)^5`

Answer

To find the probability that the product of the numbers appearing on the top faces of six rolled dice is prime, we can follow these steps:

### Step 1: Understand the possible outcomes of a single die roll
A single die can show one of the numbers: 1, 2, 3, 4, 5, or 6. Among these, the prime numbers are 2, 3, and 5.

**Hint:** Remember that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

### Step 2: Identify the conditions for the product to be prime
For the product of the numbers on the six dice to be prime, it must be the case that:
- Five of the dice must show the number 1 (since 1 is not prime and does not affect the product).
- One die must show a prime number (2, 3, or 5).

**Hint:** A prime product can only occur if there is exactly one prime factor and all other factors are 1.

### Step 3: Calculate the number of favorable outcomes
1. Choose which of the six dice will show the prime number. There are 6 ways to choose this die.
2. The chosen die can show one of the three prime numbers (2, 3, or 5).
3. The other five dice must all show 1.

Thus, the number of favorable outcomes is:
\[ 6 \text{ (ways to choose the die)} \times 3 \text{ (prime numbers)} = 18 \]

**Hint:** Think about combinations and how many ways you can arrange the outcomes.

### Step 4: Calculate the total number of outcomes
Each die has 6 faces, so the total number of outcomes when rolling six dice is:
\[ 6^6 \]

**Hint:** Use the multiplication principle of counting for independent events.

### Step 5: Calculate the probability
The probability \( P \) that the product of the numbers appearing on the top faces is prime is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{18}{6^6} \]

### Step 6: Simplify the probability
Calculating \( 6^6 \):
\[ 6^6 = 46656 \]
Thus, the probability becomes:
\[ P = \frac{18}{46656} = \frac{1}{2592} \]

### Final Answer
The probability that the product of the numbers appearing on the top faces is prime is:
\[ \frac{1}{2592} \]

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