Consider randomly selecting a student at a large university,and let A be the eve

Question

Consider randomly selecting a student at a large university,and let A be the event that the selected student has a Visa card and B be the analogous event for a MasterCard. Suppose that P (A) = .6 and P (B) = .4

1) Could it be the case that P (A B) = 0.5? Why or why not?

2) From now on, suppose that P (A B) = 0.3. What is the probability that the selected student has at least one of these types of cards?

3) What is the probability that the selected student has neither type of card?

4) Describe, in terms of A and B, the event that the selected student has a visa card but not a MasterCard, and then calculate the probability of this event?

5) Calculate the probability that the selected student has exactly one of these two types of cards?

Explanation / Answer

1. P (A B) cannot be greater than P(A) or greater than P(B). This is because P(A and B) technically talks about both A and B occuring, so it cannot be greater than probability of A occuring on its own or B occuring on its own.

Here,

P (A B) > P(B)

So, P (A B) cannot be 0.5

2. P(at least one of these cards) = P(A U B) = P(A) + P(B) - P (A B) = 0.6 + 0.4 - 0.3 = 0.7

3. P(neither A nor B) = 1 - P(A U B) = 1 - 0.7 = 0.3

4. P(Has visa but not mastercard) = P(A BC ) = P(A) - P(A B) = 0.6 - 0.3 = 0.3

5. P(excatly one card) = P(A BC ) + P(AC B ) = 0.3 + P(B) - P(A B) = 0.3 + 0.4 - 0.3 = 0.4

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