Let's clarify why we focused only on the prime numbers and why we cannot use the calculation $\frac{3}{6} \times \frac{3}{5} \times \frac{4}{6}$ for this specific probability question.

1. Understanding the Given Probability Problem:

The problem asks for the probability that:

1. We roll a prime number on both dice.
2. The sum of those two prime numbers is an even number.

2. Why Focus Only on Prime Numbers?

To solve this problem, we only consider outcomes where both dice roll prime numbers because:

• Prime numbers on a 6-sided die: 2, 3, 5 (3 prime numbers)
• Prime numbers on a 5-sided die: 2, 3, 5 (3 prime numbers)

If we want to compute the probability of getting prime numbers on both dice, we are strictly interested in the combinations where each die shows one of its prime numbers.

We then calculate the probability of these outcomes adding up to an even number, which narrows down our focus further to the combinations that meet both criteria (both prime and sum even).

3. Why Not Use $\frac{3}{6} \times \frac{3}{5} \times \frac{4}{6}$?

Let's analyze why the formula $\frac{3}{6} \times \frac{3}{5} \times \frac{4}{6}$ is incorrect for this scenario:

• $\frac{3}{6}$: Probability of rolling a prime number on a 6-sided die.
• $\frac{3}{5}$: Probability of rolling a prime number on a 5-sided die.
• $\frac{4}{6}$: Probability of getting an even total (irrelevant to the number of sides of the dice or the numbers themselves).

However, using $\frac{4}{6}$ for the probability of getting an even number is incorrect here because:

• $\frac{4}{6}$ represents the probability of rolling an even number on a 6-sided die.
• We are not interested in rolling an even number; we are interested in the total sum being even after confirming both dice rolled primes.

We cannot combine these probabilities directly because:

• The events are not independent in the way the formula suggests.
• We are not simply multiplying probabilities of independent outcomes, but instead counting specific combinations under constrained conditions (both dice must show primes, and their sum must be even).

4. Correct Approach:

Given we only care about cases where both dice show prime numbers and their sum is even, we:

1. Count only the prime number outcomes.
2. Calculate which combinations of these prime number outcomes yield an even sum.
3. Use these counts to determine the correct probability.

In summary, the correct probability of rolling a prime number on both dice and the total being even is based on counting the relevant outcomes, not multiplying separate probabilities.

Conclusion

• The correct method involves understanding the conditions and counting outcomes that match those conditions, not just multiplying probabilities independently.
• The formula you suggested doesn’t align with the conditions because the third term $\frac{4}{6}$ doesn't reflect the sum condition correctly.

Would you like more clarification, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How would the probability change if one die was an 8-sided die?
2. What is the probability of rolling two numbers that are both prime on a 6-sided die?
3. How do you calculate the probability of rolling doubles on two dice?
4. If one die shows a non-prime, what is the probability the other shows a prime?
5. What is the probability of rolling two dice and getting a sum less than 5?

Tip: Always ensure that your probabilities reflect the specific events or conditions given in a problem statement.