Consider randomly selecting a student at a certain university, and let A denote the event that
the selected individual has a Visa credit card and B be the analogous event for a MasterCard.
Suppose that P(A) = 0.6 and P(B) = 0.4
a) Could it be the case that P(A n B) = 0.5? Why or why not?
I'm not sure how to answer it or prove why it would be true or not.
however, if the notation P(A n B) means P(A conjunction B) or P(A AND B) then the answer is different.
P(A AND B) = 0.5 could not happen given your numbers above because that would imply P(B) >= 0.5.
In words: you can't have the situation where P(B) = 0.4 and P(A AND B) = 0.5, because a student possessing a MC occurs in EVERY instance of P(A AND B), but that's contradictory to P)B) = 0.4.
Clear?
In one statement you would be saying that P(B) = 0.4 and in the other P(B) >= 0.5, which is contradictory.
assuming that P(A n B) means "Probability of A and not B" or P(A AND (NOT B))
a) it should be possible to have P(A n B) = 0.5 because included in P(A) = 0.6 could be instances of randomly selecting a student who has BOTH Visa and MC. So in that case
P(A AND (NOT B)) + P(A AND B) = P(A)
OR . . . 0.5 + 0.1 = 0.6
in other words, the likelihood of a random student having BOTH cards is then P(A AND B) = 0.1.
To determine whether it is possible for P(A n B) to be 0.5, we can use the formula for the intersection of two events:
P(A n B) = P(A) * P(B|A)
where P(B|A) denotes the probability of event B occurring given that event A has already occurred.
Given that the probability of randomly selecting a student with a Visa credit card, P(A), is 0.6, and the probability of randomly selecting a student with a MasterCard, P(B), is 0.4, we need to determine whether it is possible for the product of these probabilities (0.6 * 0.4 = 0.24) to equal 0.5.
Since a probability cannot exceed 1, it is not possible for P(A n B) to be equal to 0.5. Therefore, it is not possible for the intersection of events A and B to have a probability of 0.5.
To determine whether it is possible for P(A ∩ B) to be 0.5, we need to understand the concept of independent events and conditional probability.
Two events A and B are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the occurrence of the other event. In this case, it means that having a Visa credit card (A) does not influence the probability of having a MasterCard (B), and vice versa.
To calculate the probability of two independent events occurring simultaneously, we can use the formula:
P(A ∩ B) = P(A) * P(B)
Using the given probabilities, we can calculate:
P(A ∩ B) = P(A) * P(B)
P(A ∩ B) = 0.6 * 0.4
P(A ∩ B) = 0.24
Therefore, it is not possible for P(A ∩ B) to be 0.5 because the product of the individual probabilities P(A) and P(B) does not equal 0.5.
In conclusion, the probability of both events A and B occurring simultaneously (P(A ∩ B)) cannot be 0.5 in this scenario.