Question

if ( a ) and ( b ) are events with ( p(a) = 0.4 ), ( p(a \text{ or } b) = 0.89 ), ( p(a \text{ and } b) = 0.01 ), find ( p(b) ).

Explanation:

Step1: Recall the addition rule for probabilities

The addition rule for two events \( A \) and \( B \) is given by \( P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \).

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 \( P(A) = 0.4 \), \( P(A \text{ OR } B) = 0.89 \), and \( P(A \text{ AND } B) = 0.01 \). Substituting these values into the formula, we get:
\( P(B) = 0.89 - 0.4 + 0.01 \)

Step4: Calculate the result

First, subtract \( 0.4 \) from \( 0.89 \): \( 0.89 - 0.4 = 0.49 \). Then, add \( 0.01 \) to the result: \( 0.49 + 0.01 = 0.5 \).

Answer:

\( 0.5 \)