To solve the problems involving events A and B, we will use the properties of probability.
A) Compute \( P(A \text{ and } B) \)
We use the formula for the probability of the union of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
From the information given:
- \( P(A) = 0.4 \)
- \( P(B) = 0.2 \)
- \( P(A \text{ or } B) = 0.6 \)
Plugging in these values into the union formula:
\[ 0.6 = 0.4 + 0.2 - P(A \text{ and } B) \]
Simplifying this equation:
\[ 0.6 = 0.6 - P(A \text{ and } B) \]
We can rearrange it to find \( P(A \text{ and } B) \):
\[ P(A \text{ and } B) = 0.6 - 0.6 = 0 \]
Thus,
\[ \boxed{0} \]
B) Are A and B mutually exclusive? Explain
Two events A and B are mutually exclusive if they cannot occur at the same time, which means:
\[ P(A \text{ and } B) = 0 \]
Since we calculated \( P(A \text{ and } B) = 0 \), A and B are mutually exclusive.
Therefore, the answer is
\[ \text{Yes, A and B are mutually exclusive.} \]
C) Are A and B independent? Explain
Two events A and B are independent if the occurrence of one event does not affect the occurrence of the other. This can be expressed mathematically as:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
We already found \( P(A \text{ and } B) = 0 \). Now we compute \( P(A) \times P(B) \):
\[ P(A) \times P(B) = 0.4 \times 0.2 = 0.08 \]
Since \( P(A \text{ and } B) = 0 \) is not equal to \( 0.08 \), the events A and B are not independent.
Therefore, the answer is
\[ \text{No, A and B are not independent.} \]
In summary:
- \( P(A \text{ and } B) = 0 \)
- A and B are mutually exclusive
- A and B are not independent
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