Two dice are thrown simultaneously. Find the probability that :
the product of the numbers on the dice is 8.

Answer

To find the probability that the product of the numbers on two dice is 8, we will follow these steps:

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

### Step 2: Identify Favorable Outcomes
Next, we need to find the pairs of numbers (one from each die) whose product is 8. The possible pairs are:
- (2, 4)
- (4, 2)

We can verify this by calculating the product for each pair:
- For (2, 4): \( 2 \times 4 = 8 \)
- For (4, 2): \( 4 \times 2 = 8 \)

These are the only combinations that give us a product of 8.

### Step 3: Count the Favorable Outcomes
From the above, we have identified 2 favorable outcomes:
1. (2, 4)
2. (4, 2)

### Step 4: Calculate the Probability
The probability \( P \) of an event is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Substituting the values we found:
\[ P = \frac{2}{36} \]
This simplifies to:
\[ P = \frac{1}{18} \]

### Final Answer
The probability that the product of the numbers on the dice is 8 is:
\[ \frac{1}{18} \]

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