The probability for event A is 0.4, the probability for event B is 0.2, and the probability of events A and B is 0.1.
Why are the events are not independent?
A. The sum of P(A) and P(B) is greater than P(A and B).
B. The product of P(A) and P(B) is greater than P(A and B).
C. The product of P(A) and P(B) is not equal to P(A and B).
D. The sum of P(A) and P(B) is not equal to P(A and B).
The Correct Answer and Explanation is:
The correct answer is C: The product of P(A) and P(B) is not equal to P(A and B).
Explanation:
To determine whether two events, A and B, are independent, we need to check if the following condition holds:
[
P(A \cap B) = P(A) \times P(B)
]
Where:
- ( P(A \cap B) ) represents the probability that both events A and B occur (the intersection of A and B).
- ( P(A) ) and ( P(B) ) represent the probabilities of event A and event B occurring independently.
In the problem, we are given the following probabilities:
- ( P(A) = 0.4 )
- ( P(B) = 0.2 )
- ( P(A \cap B) = 0.1 )
Step 1: Check if events are independent
To check if A and B are independent, we calculate the product of their individual probabilities:
[
P(A) \times P(B) = 0.4 \times 0.2 = 0.08
]
Next, we compare this product to the given probability of both events occurring together, ( P(A \cap B) = 0.1 ).
Since:
[
P(A \cap B) = 0.1 \quad \text{and} \quad P(A) \times P(B) = 0.08
]
We can see that:
[
P(A \cap B) \neq P(A) \times P(B)
]
Step 2: Conclusion
Because the product of ( P(A) ) and ( P(B) ) is not equal to the probability of both events occurring together, the events A and B are not independent.
For independent events, the product of their individual probabilities should equal the probability of both events occurring together. Since this condition is not met, the events are dependent, meaning that the occurrence of one event affects the probability of the other event occurring.
Thus, the correct answer is C: The product of P(A) and P(B) is not equal to P(A and B).