A and B are two events, such that `P(A)=(3)/(5) and P(B)=(2)/(3)` if A and B are independent then find P(A intersection B)

Answer

To solve the problem, we need to find the probability of the intersection of two independent events A and B, given their individual probabilities.

**Step 1: Understand the given information**
- We are given:
- \( P(A) = \frac{3}{5} \)
- \( P(B) = \frac{2}{3} \)
- We know that events A and B are independent.

**Step 2: Use the formula for independent events**
- For two independent events A and B, the probability of their intersection is given by:
\[ P(A \cap B) = P(A) \times P(B) \]

**Step 3: Substitute the values into the formula**
- Now, we substitute the given probabilities into the formula:
\[ P(A \cap B) = P(A) \times P(B) = \frac{3}{5} \times \frac{2}{3} \]

**Step 4: Perform the multiplication**
- Multiply the fractions:
\[ P(A \cap B) = \frac{3 \times 2}{5 \times 3} = \frac{6}{15} \]

**Step 5: Simplify the fraction**
- Simplifying \( \frac{6}{15} \):
\[ P(A \cap B) = \frac{2}{5} \]

**Final Answer:**
- Therefore, the probability of the intersection of events A and B is:
\[ P(A \cap B) = \frac{2}{5} \]