It looks like you're working through a probability problem involving two events, $A$ and $B$. From what you've provided, I'll clarify the steps to solve for $P(A \mid B)$ using the given information. Let’s summarize and solve it step by step:

Given:

• $P(A \cap B) = 1/4$
• $P(A) = 3/4$
• Find $P(A \mid B)$

Solution:

1. Use the formula for conditional probability: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ We need $P(A \cap B)$ and $P(B)$.

2. Calculate $P(B)$ using the formula for the union of two events: The formula is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ However, since $P(A \cup B)$ isn't directly given, we'll need to rearrange this formula to solve for $P(B)$: $P(B) = P(A \cup B) - P(A) + P(A \cap B)$ Let’s assume $P(A \cup B) = 1$ (as typically considered in a complete probability space): $P(B) = 1 - \frac{3}{4} + \frac{1}{4} = \frac{1}{2}$

3. Substitute into the conditional probability formula: With $P(A \cap B) = \frac{1}{4}$ and $P(B) = \frac{1}{2}$: $P(A \mid B) = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \cdot \frac{2}{1} = \frac{1}{2}$

Final Answer:

$P(A \mid B) = \frac{1}{2}$

Would you like a detailed explanation or have any questions about this solution?

Related Questions:

1. How do you find $P(B \mid A)$ given $P(A)$ and $P(A \cap B)$?
2. What is the significance of $P(A \cap B)$ in probability?
3. How does the formula for conditional probability change if the events are independent?
4. What is the formula for the probability of the union of two events, and when is it used?
5. How can you determine if two events $A$ and $B$ are mutually exclusive?

Tip:

Always verify if the events are independent or dependent, as this affects how probabilities are combined. For independent events, $P(A \cap B) = P(A) \cdot P(B)$.