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

Explanation

To find the minimum value of \( P(A \cap B) \) given \( P(A) = 0.8 \) and \( P(B) = 0.7 \), we can use the formula for the probability of the union of two events:

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

### Step 1: Write the formula for the union of two events
We know that the probability of any event cannot exceed 1. Therefore, we can write:

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

### Step 2: Substitute the known probabilities into the formula
Substituting the values of \( P(A) \) and \( P(B) \) into the formula, we get:

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

Substituting the values:

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

### Step 3: Simplify the equation
Now, simplify the left-hand side:

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

### Step 4: Rearrange to find \( P(A \cap B) \)
Rearranging the equation gives:

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

This simplifies to:

\[ -P(A \cap B) \leq -0.5 \]

Multiplying through by -1 (and reversing the inequality):

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

### Step 5: Conclusion
Thus, the minimum value of \( P(A \cap B) \) is:

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

So, the minimum value of \( P(A \cap B) \) is \( 0.5 \).

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

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