We are asked to find the conditional probability $P(A | B)$, which is the probability of event $A$ occurring given that event $B$ has occurred. To compute this, we use the formula:

$P(A | B) = \frac{P(A \cap B)}{P(B)}$

Step 1: Define events

• Event A: The first roll is a 3 or 4, and the second roll is an even number. The possible even numbers are 2, 4, and 6.

  So, event $A$ consists of the following pairs for the dice rolls:

  • (3, 2), (3, 4), (3, 6)
  • (4, 2), (4, 4), (4, 6)

  Therefore, $A$ contains 6 outcomes:
  $A = \{ (3, 2), (3, 4), (3, 6), (4, 2), (4, 4), (4, 6) \}$

• Event B: The sum of the two rolls is at most 7. The possible outcomes are:

  • Sum = 2: (1, 1)
  • Sum = 3: (1, 2), (2, 1)
  • Sum = 4: (1, 3), (2, 2), (3, 1)
  • Sum = 5: (1, 4), (2, 3), (3, 2), (4, 1)
  • Sum = 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
  • Sum = 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)

  Event $B$ consists of 21 outcomes.

Step 2: Calculate $P(B)$

The total number of possible outcomes when rolling two dice is $6 \times 6 = 36$. Event $B$ has 21 favorable outcomes. Thus, the probability of $B$ is:

$P(B) = \frac{21}{36} = \frac{7}{12}$

Step 3: Calculate $P(A \cap B)$

Next, we need to find the outcomes where both $A$ and $B$ occur. These are the outcomes where:

• The first roll is a 3 or 4.
• The second roll is an even number (2, 4, or 6).
• The sum of the rolls is at most 7.

We can list the outcomes from $A$ and check which ones satisfy the condition of $B$ (sum ≤ 7):

• (3, 2): sum = 5 (valid)
• (3, 4): sum = 7 (valid)
• (3, 6): sum = 9 (not valid)
• (4, 2): sum = 6 (valid)
• (4, 4): sum = 8 (not valid)
• (4, 6): sum = 10 (not valid)

Thus, the outcomes in $A \cap B$ are:
$\{ (3, 2), (3, 4), (4, 2) \}$

So, $A \cap B$ contains 3 outcomes. Therefore, the probability of $A \cap B$ is:

$P(A \cap B) = \frac{3}{36} = \frac{1}{12}$

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

Finally, we can use the formula for conditional probability:

$P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{12}}{\frac{7}{12}} = \frac{1}{7}$

Final Answer:

$P(A | B) = \frac{1}{7}$

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Would you like further details or clarification on any part of the process? Here's a list of related questions to explore:

1. What is the probability of event $A$ occurring without the condition $B$?
2. How does the conditional probability formula work in general, and why is it useful in probability theory?
3. What is the probability of the sum being exactly 7 when rolling two dice?
4. How would the probabilities change if the second roll needed to be an odd number instead of an even number?
5. How do you calculate joint probabilities for independent events?

Tip: Always remember that the total number of outcomes in any probability calculation depends on the number of independent events. In the case of two dice, it's $6 \times 6 = 36$.