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To solve this problem, we need to calculate the probability that the product of the numbers shown on two thrown dice equals 6.

### Step 1: Total Possible Outcomes
When two dice are thrown, each die has 6 faces, so the total number of possible outcomes is:
\[
6 \times 6 = 36
\]
These are all the possible combinations of numbers that can appear on the two dice.

### Step 2: Identifying the Pairs That Result in a Product of 6
We now identify all possible pairs \((x, y)\), where \(x\) is the number on the first die and \(y\) is the number on the second die, such that the product \(x \times y = 6\).

The pairs are:
- \((1, 6)\): \(1 \times 6 = 6\)
- \((2, 3)\): \(2 \times 3 = 6\)
- \((3, 2)\): \(3 \times 2 = 6\)
- \((6, 1)\): \(6 \times 1 = 6\)

Thus, there are **4 favorable outcomes**.

### Step 3: Calculating the Probability
The probability is given by the ratio of favorable outcomes to total outcomes:
\[
P(\text{product is 6}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{36} = \frac{1}{9}
\]

### Final Answer:
The probability that the product is 6 is \( \frac{1}{9} \).

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Would you like more details on any part of this solution, or do you have further questions?

Here are 5 related questions you might want to explore:
1. What is the probability of the product being 12 when two dice are thrown?
2. How many possible outcomes give a product of 4?
3. What is the probability that the product of the two dice is a prime number?
4. How would the probability change if the dice were biased?
5. What is the probability that the sum of the numbers is 6?

**Tip:** Always break down probability problems by identifying favorable outcomes and total outcomes clearly.

Two dice are thrown and the product of the two numbers shown is calculated. Find the probability that the product is 6.

Learn how to calculate the probability that the product of the numbers shown on two thrown dice equals 6. This problem covers key concepts in probability and combinatorics, including identifying favorable outcomes and calculating the total number of outcomes. Suitable for students in Grades 7-9.

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