For two events A and B, P (A cap B^ prime )= 1/5 P (A^ prime cap B)= 3/20 (A^ prime cap B^ prime )= 1/10 Find the value of P (A/B)

We are given the following probabilities for two events \( A \) and \( B \):

- \( P(A \cap B') = \frac{1}{5} \)
- \( P(A' \cap B) = \frac{3}{20} \)
- \( P(A' \cap B') = \frac{1}{10} \)

We need to find the value of \( P(A \mid B) \).

### Step 1: Find \( P(B) \)
First, we calculate \( P(B) \) using the law of total probability:

\[
P(B) = P(A \cap B) + P(A' \cap B)
\]

But we don't have \( P(A \cap B) \). We can instead calculate \( P(B') \) and then use it to find \( P(B) \).

\[
P(B') = P(A \cap B') + P(A' \cap B')
\]

Substituting the given values:

\[
P(B') = \frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10}
\]

Since \( P(B) + P(B') = 1 \):

\[
P(B) = 1 - \frac{3}{10} = \frac{7}{10}
\]

### Step 2: Find \( P(A \cap B) \)
We know:

\[
P(A) = P(A \cap B) + P(A \cap B')
\]

But, we don't need \( P(A) \) here. We can find \( P(A \cap B) \) by using \( P(B) \):

\[
P(B) = P(A \cap B) + P(A' \cap B) = \frac{7}{10}
\]

Given that \( P(A' \cap B) = \frac{3}{20} \):

\[
P(A \cap B) = \frac{7}{10} - \frac{3}{20} = \frac{14}{20} - \frac{3}{20} = \frac{11}{20}
\]

### Step 3: Calculate \( P(A \mid B) \)
The conditional probability \( P(A \mid B) \) is given by:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Substitute the values:

\[
P(A \mid B) = \frac{\frac{11}{20}}{\frac{7}{10}} = \frac{11}{20} \times \frac{10}{7} = \frac{110}{140} = \frac{11}{14}
\]

So, the value of \( P(A \mid B) \) is \( \frac{11}{14} \).

Would you like to go over the steps in more detail or have any questions about the process?

Here are five questions that expand on this topic:
1. How would the solution change if \( P(A \cap B') \) was different?
2. What is the significance of the law of total probability in this problem?
3. Can you generalize the approach to finding \( P(A \mid B) \) for any two events \( A \) and \( B \)?
4. How would you calculate \( P(B \mid A) \) using the given information?
5. What is the interpretation of the conditional probability \( P(A \mid B) \) in real-world scenarios?

**Tip:** When dealing with probabilities, drawing a Venn diagram can be helpful in visualizing the relationships between events.

Explore how to calculate conditional probability P(A given B) using given probabilities of events A and B. Includes step-by-step solution and detailed explanation. Suitable for undergraduate level and anyone interested in probability theory.

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