Given two events A and B such that (A/B) = 0.25 and P(A ∩ B) = 0.12. The value P(A ∩ B') is _

We are given:

- \( P(A | B) = 0.25 \) (i.e., the conditional probability of \( A \) given \( B \))
- \( P(A \cap B) = 0.12 \) (i.e., the joint probability of \( A \) and \( B \))

We are asked to find \( P(A \cap B') \), which is the probability that \( A \) occurs and \( B \) does not.

### Step 1: Recall the formula for conditional probability
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
We can rearrange this to solve for \( P(B) \):
\[
P(B) = \frac{P(A \cap B)}{P(A | B)} = \frac{0.12}{0.25} = 0.48
\]

### Step 2: Use the complement rule
Since \( B' \) is the complement of \( B \), we know that:
\[
P(B') = 1 - P(B) = 1 - 0.48 = 0.52
\]

### Step 3: Use the total probability rule
The total probability of \( A \) can be split as:
\[
P(A) = P(A \cap B) + P(A \cap B')
\]
We know that \( P(A \cap B) = 0.12 \), so we need to find \( P(A \cap B') \).

### Step 4: Find \( P(A) \)
To find \( P(A) \), we can express it using the law of total probability:
\[
P(A) = P(A | B) P(B) + P(A | B') P(B')
\]
Unfortunately, we do not have enough information to directly calculate \( P(A | B') \), but with the information provided, \( P(A \cap B') \) must satisfy the relation involving \( P(A) \). Would you like me to solve it in terms of \( P(A | B') \)?

---

Would you like to clarify any of the above steps or proceed further? Here are some related questions for practice:

1. What is the relationship between conditional probability and joint probability?
2. How do you calculate the complement of an event in probability?
3. What is the total probability rule and how is it used?
4. Can you explain the meaning of \( P(A \cap B) \) in the context of probability theory?
5. How do you find the probability of the complement of an event?

**Tip:** Conditional probability always requires knowing the relationship between the events involved, and it can often be calculated using joint probabilities.

Explore how to calculate P(A ∩ B') when given P(A | B) = 0.25 and P(A ∩ B) = 0.12. This problem covers key probability concepts such as conditional probability, complementary events, and the total probability rule. Suitable for students in Grades 10-12.

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