Question

If \( P(A) = P(B) = \frac{5}{13} \) and \( P(A \cap B) = \frac{2}{5} \), then find \( P(A \cup B) \).

Answer 1

Given the probabilities:

\[ P(A) = P(B) = \frac{5}{13}, \quad P(A \cap B) = \frac{2}{5}, \] we need to find \( P(A \cup B) \).

Solution:

Using the formula for the union of two events, we have:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting the given values: \[ P(A \cup B) = \frac{5}{13} + \frac{5}{13} - \frac{2}{5} \]

Now, simplify:

\[ P(A \cup B) = \frac{10}{13} - \frac{2}{5} \] To subtract these fractions, find a common denominator (the LCM of 13 and 5 is 65): \[ P(A \cup B) = \frac{50}{65} - \frac{26}{65} = \frac{24}{65} \]

Final Answer:

\[ P(A \cup B) = \frac{24}{65} \]