To calculate \( P(A \text{ and } B) \), we use the formula related to the probabilities of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
From the problem:
- \( P(A) = 0.6 \)
- \( P(B) = 0.4 \)
- \( P(A \text{ or } B) = 0.7 \)
We can substitute the values into the formula to find \( P(A \text{ and } B) \):
\[ 0.7 = 0.6 + 0.4 - P(A \text{ and } B) \]
Now, solving for \( P(A \text{ and } B) \):
\[ 0.7 = 1.0 - P(A \text{ and } B) \]
\[ P(A \text{ and } B) = 1.0 - 0.7 = 0.3 \]
Summary of Results:
- \( P(A \text{ and } B) = 0.3 \)
Are A and B Mutually Exclusive?
Two events \( A \) and \( B \) are mutually exclusive if they cannot happen at the same time, which mathematically means \( P(A \text{ and } B) = 0 \).
In this case, we found \( P(A \text{ and } B) = 0.3 \), which indicates that both events can occur simultaneously, and therefore:
- A and B are not mutually exclusive.
Are A and B Independent?
Two events \( A \) and \( B \) are independent if the occurrence of one does not affect the occurrence of the other, mathematically represented as:
\[ P(A \text{ and } B) = P(A) \cdot P(B) \]
Let's compute \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = 0.6 \cdot 0.4 = 0.24 \]
Now compare this with \( P(A \text{ and } B) \):
- \( P(A \text{ and } B) = 0.3 \)
- \( P(A) \cdot P(B) = 0.24 \)
Since \( P(A \text{ and } B) \neq P(A) \cdot P(B) \), it implies that the occurrence of \( A \) does have an effect on the occurrence of \( B \) and vice versa:
- A and B are not independent.
Final Conclusions:
- \( P(A \text{ and } B) = 0.3 \)
- A and B are not mutually exclusive.
- A and B are not independent.