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.