Consider randomly selecting a student at a large university. Let A be the event that the selected student has a Visa card, let B be the analogous event for MasterCard, and let C be the event that the selected student has an American Express card. Suppose that P (A) = 0.6, P (B) = 0.4, and P (A ∩ B) = 0.3, suppose that P (C) = 0.2, P (A ∩ C) = 0.12, P (B ∩ C) = 0.1, and P (A ∩ B ∩ C) = 0.07. (a) What is the probability that the selected student has at least one of the three types of cards?

0.75

Step-by-step explanation:

Given,

P (A) = 0.6, P (B) = 0.4, P (C) = 0.2,

P (A ∩ B) = 0.3, P (A ∩ C) = 0.12, P (B ∩ C) = 0.1 and P (A ∩ B ∩ C) = 0.07,

Where,

A = event that the selected student has a Visa card,

B = event that the selected student has a MasterCard,

C = event that the selected student has an American Express card,

We know that,

P (A ∪ B ∪ C) = P (A) + P (B) + P (C) - P (A ∩ B) - P (A ∩ C) - P (B ∩ C) + P (A ∩ B ∩ C)

= 0.6 + 0.4 + 0.2 - 0.3 - 0.12 - 0.1 + 0.07

= 0.75

Hence, the probability that the selected student has at least one of the three types of cards is 0.75.