The problem from the image reads:

"Two fair six-sided dice are cast (one green and one red). What is the probability the red die rolled an even number, if it is known that the sum of the two dice was greater than 8?"

Solution Approach:

We will use conditional probability to solve this problem.

Let:

• $A$: The event that the red die shows an even number.
• $B$: The event that the sum of the two dice is greater than 8.

The probability we want is $P(A | B)$, which can be calculated using the formula for conditional probability: $P(A | B) = \frac{P(A \cap B)}{P(B)}$

Step 1: Identify all possible outcomes where the sum is greater than 8.

The two dice are numbered from 1 to 6, so their maximum possible sum is 12. We need the outcomes where the sum is greater than 8. These are:

• Sum = 9: $(3,6), (4,5), (5,4), (6,3)$
• Sum = 10: $(4,6), (5,5), (6,4)$
• Sum = 11: $(5,6), (6,5)$
• Sum = 12: $(6,6)$

So, the total outcomes where the sum is greater than 8 is 10 outcomes.

Step 2: Count favorable outcomes where the red die is even and the sum is greater than 8.

For this, we need to look at the pairs where the red die shows an even number (2, 4, or 6) and the sum is greater than 8:

• Red die = 6: $(6,3), (6,4), (6,5), (6,6)$ → 4 outcomes.
• Red die = 4: $(4,5), (4,6)$ → 2 outcomes.

So, there are 6 favorable outcomes where the red die is even and the sum is greater than 8.

Step 3: Calculate probabilities.

• $P(B)$: The probability that the sum is greater than 8. There are 10 favorable outcomes out of a total of 36 possible outcomes for rolling two dice, so: $P(B) = \frac{10}{36} = \frac{5}{18}$
• $P(A \cap B)$: The probability that the red die is even and the sum is greater than 8. There are 6 favorable outcomes, so: $P(A \cap B) = \frac{6}{36} = \frac{1}{6}$

Step 4: Calculate $P(A | B)$.

Using the formula for conditional probability: $P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{6}}{\frac{5}{18}} = \frac{1}{6} \times \frac{18}{5} = \frac{3}{5}$

Thus, the probability that the red die rolled an even number, given that the sum of the two dice was greater than 8, is $\frac{3}{5}$ or 60%.

Would you like more details or have any questions about this solution?

Here are 5 related questions:

1. What is the probability the sum of the two dice equals exactly 9?
2. How would the probability change if the dice were not fair?
3. What is the probability of rolling an odd number on both dice?
4. Can you calculate the probability of rolling a sum less than 5?
5. How do you calculate probabilities when more than two dice are involved?

Tip: Conditional probability is about narrowing down the possible outcomes to those that satisfy certain conditions, simplifying the problem.