Question

if ( a ) and ( b ) are events with ( p(a) = 0.5 ), ( p(b) = 0.3 ), ( p(a \text{ or } b) = 0.67 ), find ( p(a \text{ and } 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 \( P(A \text{ AND } B) \)

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

Step3: Substitute the given values

We know that \( P(A) = 0.5 \), \( P(B) = 0.3 \), and \( P(A \text{ OR } B) = 0.67 \). Substituting these values into the formula, we get:
\( P(A \text{ AND } B) = 0.5 + 0.3 - 0.67 \)

Step4: Calculate the result

First, add \( 0.5 \) and \( 0.3 \): \( 0.5 + 0.3 = 0.8 \). Then subtract \( 0.67 \) from \( 0.8 \): \( 0.8 - 0.67 = 0.13 \).

Answer:

\( 0.13 \)