Answer each problem: Probability set of notations. 2.8) Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed....

Answer each problem:
Probability set of notations.
2.8)
Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. [These pairs can be listed, for example by letting (2,3) denote that a 2 was observed on the first die and a 3 on the second.]
a) Define the following subsets of S:
A: The number on the second die is even.
B: The sum of the two numbers is even.
C: At least one number in the pair is odd.
b) List the points in A, A B, A B¯, A¯ B, and A¯ C.
Counting sample points.
2.36)
An assembly operation in a manufacturing plant requires three steps that can be performed in any sequence. How many different ways can the assembly be performed?
2.40)
A brand of automobile comes in five different styles, with four types of engines, with two types of transmissions and in eight colors.

1. How many autos would a dealer have to stock if he included one for each style-engine-transmission combination?


2. How many would a distribution center have to carry if all colors of cars were stocked for each combination in part (a)?

2.58)
Five cards are dealt from a standard 52 card deck. What is the probability that we draw

1. 3 aces and 2 kings?


2. a “full house” (3 cards of one kind, 2 cards of another kind)?

Two laws of probability.
2.92)
A policy requiring all hospitals employees to take lie detector tests may reduce losses due to theft, but some employees regard such tests as a violation of their rights. Past experience indicates that lie detectors have accuracy rates that vary from 92% to 99%. To gain some insight into the risk that employees face when taking a lie detector test, suppose that the probability is .05 that a lie detector concludes that a person is lying who, in fact, is telling the truth and suppose that any pair of tests are independent. What is the probability that a machine will conclude that

1. each of three employees is lying when all are telling the truth?


2. At least one of the three employees is lying when all are telling the truth?

Calculating probability of a event: the event-composition method.
2.110)
Of the items produced daily by a factory, 40% com from line I and 60% from line II. Line I has a defect rate of 8%, whereas line II has a defect rate of 10%. If an item is chosen at random from the day’s production, find the probability that it will not be defective.
The law of total probability and Bayes’ rule.
2.124)
A population of voters contains 40% Republicans and 60% Democrats. It is reported that 30% of the Republicans and 70% of the Democrats favor an election issue. A person chosen at random from this population is found to favor the issue in question. Find the conditional probability that this person is a Democrat.