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 MasterCard. Suppose that P(A) = 0.6 and P(B) = 0.3. (a) Could it be the case that P(A ∩ B) = 0.5? Why or why not? [Hint: For any two sets A and B if A is a subset of B then P(A) ≤ P(B).]

Answer 1

We have that
`B = 0.3`.

So

`0.3 = b + (A \cap B)`

However, b is a probability, which means that it cannot be negative. So no, P(A ∩ B) cannot be 0.5. It can, at most, be 0.3.

Explanation:

Event A:

Probability that a student has a Visa card.

Event B:

Probability that the student has a MasterCard.

We have that:

`A = a + (A \cap B)`

In which a is the probability that a student has a Visa card but not a MasterCard and
`A \cap B` is the probability that a student has both these cards.

By the same logic, we have that:

`B = b + (A \cap B)`

In this problem, we have that:

`A = 0.6, B = 0.3`

(a) Could it be the case that P(A ∩ B) = 0.5?

We have that
`B = 0.3`.

So

`0.3 = b + (A \cap B)`

However, b is a probability, which means that it cannot be negative. So no, P(A ∩ B) cannot be 0.5. It can, at most, be 0.3.