To solve for $P(A)$ and $P(B)$ given the probabilities involving the events $A$ and $B$, we can start by clarifying the notations used and what each given probability represents.

1. Given Information:

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

   Notice that the notation $B'$ represents the complement of $B$. The probability $P(A \mid B \cap B')$ is not typical because $B \cap B'$ is always an empty set (since no event can be both $B$ and $B'$ simultaneously). So, assuming this is a typo, I will interpret the problem as follows:

2. Correct Interpretation:

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

   Let $P(A)$ and $P(B)$ be the probabilities of events $A$ and $B$ respectively. Also, let $P(B') = 1 - P(B)$ and $P(A') = 1 - P(A)$.

Step 1: Express $P(A \cap B')$

$P(A \cap B') = \frac{1}{5} \times P(B')$ $P(A \cap B') = \frac{1}{5} \times (1 - P(B))$

Step 2: Use the given probabilities

We know: $P(A' \cap B) = \frac{3}{20}$ $P(A' \cap B') = \frac{1}{10}$

Step 3: Express $P(A')$

Using the fact that $P(A') = P(A' \cap B) + P(A' \cap B')$: $P(A') = \frac{3}{20} + \frac{1}{10} = \frac{3}{20} + \frac{2}{20} = \frac{5}{20} = \frac{1}{4}$ So, $P(A) = 1 - P(A') = 1 - \frac{1}{4} = \frac{3}{4}$

Step 4: Use Total Probability Formula

The total probability $P(A)$ can also be calculated as: $P(A) = P(A \cap B) + P(A \cap B')$ But since $P(A) = \frac{3}{4}$ and $P(A \cap B') = \frac{1}{5} \times (1 - P(B))$, we get: $\frac{3}{4} = P(A \cap B) + \frac{1}{5} \times (1 - P(B))$

Step 5: Solve for $P(B)$

You already know $P(A' \cap B) = \frac{3}{20}$ and $P(A') = \frac{1}{4}$, so let's solve for $P(B)$:

Let $P(B) = x$. Then: $P(A' \cap B) = P(A' \mid B) \times P(B) = \frac{3}{20}$ Given $P(A') = \frac{1}{4}$, this simplifies the equation to: $P(A' \mid B) \times x = \frac{3}{20}$

This equation, combined with the information already known, can be used to solve for $x = P(B)$.

However, it seems there is missing or inconsistent information, and further steps require assumptions.

Would you like more details or have any specific questions about a particular part of this process?

Related Questions:

1. How do you calculate $P(A \cap B)$ given the individual probabilities?
2. What is the difference between conditional probability and joint probability?
3. How can you express $P(A' \cap B')$ using complements?
4. What is the significance of total probability in probability theory?
5. How do we interpret intersections like $B \cap B'$ in probability?

Tip: Always ensure that the events are mutually exclusive before interpreting intersections like $B \cap B'$ in conditional probability contexts.