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 $\frac{1}{9}$.

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.