Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is a prime number.

Answer

To find the probability that the product of the numbers appearing on two thrown dice is a prime number, we can follow these steps:

### Step 1: Determine the total number of outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the prime numbers
The prime numbers less than or equal to 6 (the maximum number on a die) are:
- 2
- 3
- 5

### Step 3: Determine the conditions for the product to be a prime number
A product of two numbers can be a prime number if one of the numbers is 1 (since 1 multiplied by a prime number gives the prime number itself). Thus, we need to find pairs where the product results in a prime number.

### Step 4: List the favorable outcomes
1. **For the product to be 2**:
- (1, 2)
- (2, 1)
- Total = 2 outcomes

2. **For the product to be 3**:
- (1, 3)
- (3, 1)
- Total = 2 outcomes

3. **For the product to be 5**:
- (1, 5)
- (5, 1)
- Total = 2 outcomes

### Step 5: Calculate the total favorable outcomes
Now, we add the favorable outcomes:
\[ \text{Total favorable outcomes} = 2 + 2 + 2 = 6 \]

### Step 6: Calculate the probability
The probability \( P \) that the product is a prime number is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Final Answer
The probability that the product of the numbers appearing on the two dice is a prime number is:
\[ \frac{1}{6} \]

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