Events M and N have probabilities such that P(M) = 0.4, P(N) = 0.28, P(M ∪ N) = 0.56, and P(M ∩ N) = 0.18. Are events M and N independent?

Question

Question 10 options:

No, because P(M) · P(N) ≠ P(M ∩ N).

No, because P(M) - P(N) = P(M ∩ N).

Yes, because P(M) + P(N) = P(M ∪ N).

Yes, because P(M) · P(N) ≠ P(M ∪ N).

Answer 1

To determine if events M and N are independent, we need to check if the following condition holds:

\[ P(M ∩ N) = P(M) \times P(N) \]

Given:

• \( P(M) = 0.4 \)
• \( P(N) = 0.28 \)
• \( P(M ∩ N) = 0.18 \)

First, we calculate \( P(M) \times P(N) \):

\[ P(M) \times P(N) = 0.4 \times 0.28 = 0.112 \]

Now, we compare this result with \( P(M ∩ N) \):

\[ P(M ∩ N) = 0.18 \]

Since \( 0.18 \neq 0.112 \), we conclude that:

\[ P(M ∩ N) \neq P(M) \times P(N) \]

Therefore, the events M and N are not independent.

The correct answer is:

No, because \( P(M) \cdot P(N) \neq P(M ∩ N) \).