Final answer:
To calculate P(B ∪ D), add the probabilities of event B and event D and subtract the probability of their intersection. To calculate P(A ∩ C), multiply the probabilities of event A and event C. B ∪ D represents either being under 21 or female and A ∩ C represent being aged 21 and over and male. To find the probability of the selected person being female given they are under 21, use the conditional probability formula P(D|B).
Step-by-step explanation:
To calculate P(B ∪ D), we need to find the probability of either event B or event D happening. Event B represents the person being under 21, and event D represents the person being female. To find the probability, we add the probabilities of B and D and subtract the probability of their intersection. P(B ∪ D) = P(B) + P(D) - P(B ∩ D).
To calculate P(A ∩ C), we need to find the probability of both event A and event C happening. Event A represents the person being aged 21 and over, and event C represents the person being male. To find the probability, we multiply the probabilities of A and C. P(A ∩ C) = P(A) * P(C).
In words, B ∪ D represents the event of selecting a person who is either under 21 or female. A ∩ C represents the event of selecting a person who is both aged 21 and over and male.
To find the probability that the selected person is female given that the selected person is under 21, we use the conditional probability formula P(D|B) = P(D ∩ B) / P(B). We already have the probabilities for D ∩ B and P(B) from the previous calculations.