Question

Given events \( A \) and \( B \) are absolute and \( P(A) = p \), \( P(B) = \frac{1{2} \), and \( P(A \cup B) = \frac{3}{5} \), the value of \( p \) is __________.}

Hint:

For the union of two events, use the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), and remember that if the events are absolute (i.e., no overlap), \( P(A \cap B) = 0 \).

Answer 1

Step 1: We are given the following probabilities: \[ P(A) = p, \quad P(B) = \frac{1}{2}, \quad P(A \cup B) = \frac{3}{5}. \] The formula for the probability of the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] Step 2: Substitute the given values into the formula: \[ \frac{3}{5} = p + \frac{1}{2} - P(A \cap B). \] Step 3: Since events \( A \) and \( B \) are absolute, \( P(A \cap B) = 0 \), as they do not overlap. Thus, the equation becomes: \[ \frac{3}{5} = p + \frac{1}{2}. \] Step 4: Solving for \( p \): \[ p = \frac{3}{5} - \frac{1}{2}. \] To subtract the fractions, find a common denominator: \[ p = \frac{6}{10} - \frac{5}{10} = \frac{1}{10}. \] Thus, the value of \( p \) is \( \frac{1}{5} \).