Mathematics 375 – Probability and Statistics 1 Whole Class Discussion Problems – The “Event-Composition Method” September 16, 2011

Background

The idea of the event-composition method is to recognize that the event of interest E can be defined using some combination of unions, intersections, or complements of other known events. Then apply some combination of the additive and multiplicative laws of probability:

P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

and

P (A ∩ B) = P (A)P (B|A) = P (B)P (A|B)

and the rule for complements:

P (\overline{A}) = 1 − P (A)

to find P (E) from the other given information.

Discussion Questions

A) Given: P (A) = .2, P (B) = .3, P (A ∪ B) = .4. Determine:

1) P (\overline{A} ∪ \overline{B}).

2) P (\overline{A}|B).

B) The circles marked 1,2,3,4 in the following diagrams represent electrical relays that operate independently and function properly with probability p = .85.

Which design yields the higher probability that current will flow when the relays are activated?

C) An accident victim will die unless he receives type A+ blood (strictly) before 10 minutes elapse. He will be saved if he does get the blood transfusion. Potential donors and a reusable blood typing kit are available, but it takes 2 minutes to determine each possible donor’s blood type, and only 40% of them have type A+ blood. What is the probability that the victim will be saved if only one donor’s blood can be typed at a time? (Follow-up: Are you making an assumption in your solution?)