To solve the problems, we can utilize the formulas related to probability.
A) Compute \( P(A \text{ and } B) \)
We know 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:
- \( P(A) = 0.3 \)
- \( P(B) = 0.7 \)
- \( P(A \text{ or } B) = 0.9 \)
Substituting the known values into the union formula gives:
\[ 0.9 = 0.3 + 0.7 - P(A \text{ and } B) \]
Simplifying this equation:
\[ 0.9 = 1.0 - P(A \text{ and } B) \]
To isolate \( P(A \text{ and } B) \):
\[ P(A \text{ and } B) = 1.0 - 0.9 = 0.1 \]
So,
\[ \boxed{P(A \text{ and } B) = 0.1} \]
B) Are A and B mutually exclusive?
Two events are mutually exclusive if they cannot occur at the same time, which means:
\[ P(A \text{ and } B) = 0 \]
Since we calculated \( P(A \text{ and } B) = 0.1 \), which is greater than 0, this indicates that events A and B can occur together. Thus, A and B are not mutually exclusive.
C) Are A and B independent?
Two events are independent if the occurrence of one does not affect the occurrence of the other, which mathematically can be expressed as:
\[ P(A \text{ and } B) = P(A) \cdot P(B) \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = 0.3 \cdot 0.7 = 0.21 \]
Now we compare \( P(A \text{ and } B) \) with \( P(A) \cdot P(B) \):
- \( P(A \text{ and } B) = 0.1 \)
- \( P(A) \cdot P(B) = 0.21 \)
Since \( P(A \text{ and } B) = 0.1 \) is not equal to \( P(A) \cdot P(B) = 0.21 \), A and B are not independent.
Summary:
- A) \( P(A \text{ and } B) = 0.1 \)
- B) A and B are not mutually exclusive.
- C) A and B are not independent.