P(A) = 0.31, P(B) = 0.44, and P(AandB) = 0.25. What is P(B | notA )?

To find \( P(B \mid \neg A) \), we can use the following formula for conditional probability:

\[
P(B \mid \neg A) = \frac{P(B \text{ and } \neg A)}{P(\neg A)}
\]

### Step 1: Calculate \( P(\neg A) \)
\[
P(\neg A) = 1 - P(A) = 1 - 0.31 = 0.69
\]

### Step 2: Calculate \( P(B \text{ and } \neg A) \)
We know that:
\[
P(B \text{ and } \neg A) = P(B) - P(A \text{ and } B)
\]

Given:
- \( P(B) = 0.44 \)
- \( P(A \text{ and } B) = 0.25 \)

Thus:
\[
P(B \text{ and } \neg A) = 0.44 - 0.25 = 0.19
\]

### Step 3: Calculate \( P(B \mid \neg A) \)
Now we can find \( P(B \mid \neg A) \):

\[
P(B \mid \neg A) = \frac{P(B \text{ and } \neg A)}{P(\neg A)} = \frac{0.19}{0.69} \approx 0.2754
\]

So, the conditional probability \( P(B \mid \neg A) \) is approximately \( 0.2754 \).

Would you like further details or explanations? Here are some related questions to consider:

1. How would the result change if \( P(A \text{ and } B) \) was different?
2. Can you explain how the conditional probability formula works for different cases?
3. How does the relationship between \( P(A) \), \( P(B) \), and \( P(A \text{ and } B) \) impact joint probabilities?
4. What would \( P(\neg B \mid A) \) be in this scenario?
5. How would this change if events A and B were independent?

**Tip:** Always verify if the events are independent; this can greatly simplify or alter the calculations.

P(A) = 0.31, P(B) = 0.44, and P(A and B) = 0.25. What is P(B | notA)?

Find the conditional probability P(B | notA) using the values P(A) = 0.31, P(B) = 0.44, and P(A and B) = 0.25. This problem involves calculating conditional probability, using the formula P(B | notA) = P(B and notA) / P(notA). Suitable for Grades 10-12.

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