Question

question
if ( a ) and ( b ) are events with ( p(a) = 0.5 ), ( p(a \text{ or } b) = 0.65 ), ( p(a \text{ and } b) = 0.15 ), find ( p(b) ).
provide your answer below:

Explanation:

Step1: Recall the addition rule for probability

The formula for the probability of the union of two events \( A \) and \( B \) is \( P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \).

Step2: Rearrange the formula to solve for \( P(B) \)

We can rearrange the formula to isolate \( P(B) \): \( P(B) = P(A \text{ OR } B) - P(A) + P(A \text{ AND } B) \).

Step3: Substitute the given values

We know that \( P(A) = 0.5 \), \( P(A \text{ OR } B) = 0.65 \), and \( P(A \text{ AND } B) = 0.15 \). Substituting these values into the formula:
\( P(B) = 0.65 - 0.5 + 0.15 \)

Step4: Calculate the result

First, \( 0.65 - 0.5 = 0.15 \), then \( 0.15 + 0.15 = 0.3 \).

Answer:

\( 0.3 \)