Questions: If P(A or B) = 0.9, P(A) = 0.6, and P(B) = 0.5, determine P(A and B)

Transcript text: If P(A or B) = 0.9, P(A) = 0.6, and P(B) = 0.5, determine P(A and B)

Solution

Solution Steps

To determine \( P(A \text{ and } B) \), we can use the formula for the probability of the union of two events: \[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \] Given: \[ P(A \text{ or } B) = 0.9 \] \[ P(A) = 0.6 \] \[ P(B) = 0.5 \]

We can rearrange the formula to solve for \( P(A \text{ and } B) \): \[ P(A \text{ and } B) = P(A) + P(B) - P(A \text{ or } B) \]

Step 1: Given Information

We are given the following probabilities:

\( P(A \text{ or } B) = 0.9 \)
\( P(A) = 0.6 \)
\( P(B) = 0.5 \)

Step 2: Use the Formula for Union of Events

To find \( P(A \text{ and } B) \), we use the formula: \[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \] Rearranging this gives us: \[ P(A \text{ and } B) = P(A) + P(B) - P(A \text{ or } B) \]

Step 3: Substitute the Values

Substituting the known values into the equation: \[ P(A \text{ and } B) = 0.6 + 0.5 - 0.9 \]

Step 4: Calculate the Result

Calculating the right-hand side: \[ P(A \text{ and } B) = 1.1 - 0.9 = 0.2 \]