A \( P(A \mid B) \) is the probability of rolling a four or five on the first die, followed by an odd number on the second die given that the sum of these two rolls is at most eight. B \( P(A \mid B) \) is the probability that the sum of rolling two dice is at most eight given the fact that we have a four or five on the first die, followed by an odd number on the second die. C \( P(A \mid B) \) is the probability of rolling a four or five on the list die, followed by an odd number on the second die given that the sum of these two rolls is at least eight. Done of the above Find \( P(A \mid B) \). \( P(A \mid B)= \) Enter your answer

To solve for \( P(A \mid B) \) in each of the described situations, we need to begin by defining events \( A \) and \( B \): - Event \( A \): Rolling a four or five on the first die and an odd number on the second die. - Event \( B \): The condition related to the sum of the two rolls. **For scenario A and B:** 1. **Calculate \( P(A \mid B) \)** where B is the sum of the rolls at most 8: - Possible outcomes when the sum of two dice is at most 8: There are 28 valid outcomes. - Outcomes for event A (4 or 5 on the first die): - 4 on first, odd on second: (4,1), (4,3), (4,5) → 3 outcomes - 5 on first, odd on second: (5,1), (5,3) → 2 outcomes - Total valid outcomes for A under B: 3 + 2 = 5 Thus, \( P(A \mid B) = \frac{5}{28} \). **For scenario C:** 1. **Calculate \( P(A \mid B) \)** where B is the sum of the rolls at least 8: - Possible outcomes when the sum of two dice is at least 8: There are 36 - 21 = 15 valid outcomes (out of 36 total). - For event A: - With 4 on first: (4,4) → 1 outcome (4+4=8) - With 5 on first: (5,3), (5,5) → 2 outcomes Thus, \( P(A \mid B) = \frac{3}{15} = \frac{1}{5} \). Given the original question, it depends on the scenario, but since A and B describe mutually exclusive events, the answer would refer to the specifics provided, ultimately calculating \( P(A \mid B) \) as: 1. **For the scenario with the sum at most 8:** \( P(A \mid B) = \frac{5}{28} \) 2. **For the scenario with the sum at least 8:** \( P(A \mid B) = \frac{1}{5} \) Note: Be sure to choose the appropriate context for your problem as it will influence which answer is correct!