We have been provided with three probabilities:
- The probability of Event A happening, denoted as P(A), is 0.4.
- The probability of Event B happening, P(B), is also 0.4.
- The probability of Event A happening given that Event B has occurred, denoted as P(A|B), is 0.5.
To solve this question, we will need to recall some essential concepts of probability theory.
a. The formula for finding the probability of two events occurring together, A and B, known as the joint probability, is
P(A ∩ B) = P(A|B) * P(B)
Substitute the given probabilities into the equation:
P(A ∩ B) = 0.5 * 0.4 = 0.2
Therefore, the joint probability of events A and B, P(A ∩ B), is 0.2.
b. Now, we want to find the probability of Event B happening given that Event A has occurred, denoted as P(B|A). The formula for conditional probability is
P(B|A) = P(A ∩ B) / P(A)
Substitute the relevant probabilities into the equation:
P(B|A) = 0.2 / 0.4 = 0.5
Therefore, the conditional probability of event B given that event A has occurred, P(B|A), is 0.5.
So we determined that the joint probability of Event A and B is 0.2, and the conditional probability of Event B given Event A is 0.5.