If A and B are two events such that `P(A) =0.6 , P(B) = 0.2 and P(A//B) = 0.5 ` , then P(A,B)is equal to

Options

A. `1/10`
B. `3/10`
C. `3/8`
D. `6/7`

Answer

To solve the problem, we need to find \( P(A \cap B) \) given the probabilities \( P(A) = 0.6 \), \( P(B) = 0.2 \), and \( P(A|B) = 0.5 \).

### Step-by-Step Solution:

1. **Understand the relationship between conditional probability and joint probability**:
The formula for conditional probability is given by:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
This means that the probability of event A given event B is equal to the probability of both A and B occurring divided by the probability of B.

**Hint**: Remember that conditional probability relates the likelihood of one event occurring given that another event has occurred.

2. **Rearranging the formula**:
We can rearrange the formula to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A|B) \times P(B) \]

**Hint**: When rearranging formulas, ensure you maintain the equality by performing the same operation on both sides.

3. **Substituting the known values**:
Now, substitute the known values into the equation:
\[ P(A \cap B) = 0.5 \times 0.2 \]

**Hint**: Make sure to multiply the probabilities correctly, as they are decimal values.

4. **Calculating the result**:
Perform the multiplication:
\[ P(A \cap B) = 0.1 \]

**Hint**: When multiplying decimals, remember to count the number of decimal places in the factors to place the decimal point correctly in the result.

### Final Answer:
Thus, the probability \( P(A \cap B) \) is equal to \( 0.1 \).