The probabilities of two events A and B are given as P(A)=0.8 and P(B)=0.7. What is the minimum value of `P(A nn B)`?

Options

A. 0
B. 0.1
C. 0.5
D. 1

Answer

To find the minimum value of \( P(A \cap B) \), we can use the formula for the probability of the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Where:
- \( P(A) = 0.8 \)
- \( P(B) = 0.7 \)

First, we need to determine the maximum value of \( P(A \cup B) \). The probability of the union of two events cannot exceed 1. Therefore, we have:

\[ P(A \cup B) \leq 1 \]

Substituting the values we know into the union formula:

\[ 1 \geq P(A) + P(B) - P(A \cap B) \]

Substituting the given probabilities:

\[ 1 \geq 0.8 + 0.7 - P(A \cap B) \]

This simplifies to:

\[ 1 \geq 1.5 - P(A \cap B) \]

Rearranging gives:

\[ P(A \cap B) \geq 1.5 - 1 \]

Thus:

\[ P(A \cap B) \geq 0.5 \]

This means the minimum value of \( P(A \cap B) \) is \( 0.5 \).

### Final Answer:
The minimum value of \( P(A \cap B) \) is \( 0.5 \).