Mathematics 375 – Probability and Statistics 1 Computing Probabilities – Example Problems September 17, 2009

Background

Last time we introduced the conditional probability:

P (A|B) = \frac{P (A ∩ B)}{P (B)}

and several other useful facts:

1) The ”multiplicative rule”: P (A ∩ B) = P (A|B)P (B)

2) The ”additive rule”: P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

3) The ”complement rule”: P (\overline{A}) = 1 − P (A). Today we want to practice using these on the following examples.

Questions

A) (Hypothetical!) Polls show that 70% of Democrats and 20% of Republicans favor the current health care reform proposals. The population consist of 55% Democrats and 45% Republicans. A single person is chosen randomly. What is the probability that she is a Democrat, given that she favors the health care reform proposals?

B) As items come to the end of a production line, an inspector chooses which items are to go through a complete inspection. 10% of all the items are defective. 60% of the items that are defective go through a complete inspection and 20% of the items that are not defective go through a complete inspection. What is the probability that an item is defective, given that that it is chosen for a complete inspection.

C) (From the Society of Actuaries ”P Exam” Study Materials) Of 937 men who died in 1999, 210 died of causes related to heart disease. 312 of the 937 men had at least one parent who had heart disease. Of those 312, 102 died of causes related to heart disease. If a man is randomly selected from the group, what is the probability that he died of heart disease, given that neither parent had heart disease?

D) An accident victim will die unless he receives type A+ blood before 10 minutes elapse, and he will be saved if he does get the transfusion. Potential blood donors and a reusable blood typing kit are available. The donors do not know their blood types, though. It takes 2 minutes to determine each one’s type and only one can be processed ata time. Only 40% of the potential donors have type A+ blood. What is the probability that the victim will be saved?