If p(A)=.50, p(B)=.60, and p(A and B)=.30, Are A and B mutually exclusive events? and independent events?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
To determine whether events A and B are mutually exclusive and independent, we need to consider their probabilities and the relationship between them.
Mutually Exclusive Events:
Two events are considered mutually exclusive if they cannot occur at the same time. In other words, if event A occurs, event B cannot occur, and vice versa. To determine this, we check if the probability of the intersection of A and B (denoted as "p(A ∩ B)") is zero.
Given that p(A) = 0.50, p(B) = 0.60, and p(A ∩ B) = 0.30:
p(A ∩ B) = 0.30
Since the probability of A and B occurring together is not zero (p(A ∩ B) ≠ 0), events A and B are not mutually exclusive.
Independent Events:
Two events are independent if the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. In other words, if the outcome of event A is known, it does not provide any information about the occurrence of event B, and vice versa. To determine this, we check if the probability of the intersection of A and B is equal to the product of their individual probabilities:
p(A ∩ B) = p(A) * p(B)
Substituting the given values:
0.30 = 0.50 * 0.60
0.30 = 0.30
Since the equation holds true (0.30 = 0.30), events A and B are independent.
In conclusion, A and B are not mutually exclusive but are independent events.
Two events are mutually exclusive if they cannot occur at the same time. So, to determine if A and B are mutually exclusive, we need to check if p(A and B) = 0.
Given that p(A and B) = 0.30, this means that A and B can occur at the same time. Therefore, A and B are not mutually exclusive.
To determine if two events are independent, we need to check if p(A) × p(B) = p(A and B).
Given that p(A) = 0.50, p(B) = 0.60, and p(A and B) = 0.30, we can calculate:
0.50 × 0.60 = 0.30
Since p(A) × p(B) = p(A and B), A and B are independent events.
In summary, A and B are not mutually exclusive but are independent events.