IF A and B be two events in a sample space such that ` P(A ) = 2/5 ,P(B ) = 1/2 and P(A uu B ) = 3/5 ,` find `P(A nn B )`

Options

A. `3/10`
B. `7/10`
C. `4/7`
D. `4/15`

Answer

To find \( P(A \cap B) \), we can use the formula for the probability of the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Given:
- \( P(A) = \frac{2}{5} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{3}{5} \)

We can substitute these values into the formula:

\[ \frac{3}{5} = \frac{2}{5} + \frac{1}{2} - P(A \cap B) \]

Next, we need to find a common denominator to combine the fractions on the right side. The least common multiple of 5 and 2 is 10.

Converting each term to have a denominator of 10:

- \( \frac{2}{5} = \frac{4}{10} \)
- \( \frac{1}{2} = \frac{5}{10} \)
- \( \frac{3}{5} = \frac{6}{10} \)

Now, substituting these values back into the equation:

\[ \frac{6}{10} = \frac{4}{10} + \frac{5}{10} - P(A \cap B) \]

Combining the fractions on the right side:

\[ \frac{6}{10} = \frac{9}{10} - P(A \cap B) \]

Now, we can isolate \( P(A \cap B) \):

\[ P(A \cap B) = \frac{9}{10} - \frac{6}{10} \]

Calculating the right side:

\[ P(A \cap B) = \frac{3}{10} \]

Thus, the probability of the intersection of events A and B is:

\[ \boxed{\frac{3}{10}} \]