Final answer:
a. P(B|A) = 0.4615. b. P(B' ∣A) = 0.5385. c. P(A|B) = 0.8571. d. P(A ′∣B) = 0.1429. e. The probability of having a Visa Card given that the person has at least one card cannot be determined without additional information.
Step-by-step explanation:
a. To calculate P(B|A), which is the probability of event B happening given that event A has already occurred, we use the formula P(B|A) = P(A∩B) / P(A). Given that P(A∩B) = 0.30 and P(A) = 0.65, we have P(B|A) = 0.30 / 0.65 = 0.4615.
b. To calculate P(B' ∣A), which is the probability of event B' (not B) happening given that event A has already occurred, we use the formula P(B' ∣A) = 1 - P(B|A). Substituting the value of P(B|A) from part (a), we have P(B' ∣A) = 1 - 0.4615 = 0.5385.
c. To calculate P(A|B), which is the probability of event A happening given that event B has already occurred, we use the formula P(A|B) = P(A∩B) / P(B). Given that P(A∩B) = 0.30 and P(B) = 0.35, we have P(A|B) = 0.30 / 0.35 = 0.8571.
d. To calculate P(A ′∣B), which is the probability of event A' (not A) happening given that event B has already occurred, we use the formula P(A ′∣B) = 1 - P(A|B). Substituting the value of P(A|B) from part (c), we have P(A ′∣B) = 1 - 0.8571 = 0.1429.
e. To calculate the probability that the selected individual has a Visa Card given that they have at least one card, we need to find P(Visa | at least one card), which is equal to P(Visa ∩ at least one card) / P(at least one card). Without additional information, we cannot determine the values for these probabilities.