Engineering Probability & Statistics
Sharif University of Technology
Hamid R. Rabiee & S. Abbas Hosseini
October 5, 2014
CE 181
Date Due: Mehr 21st , 1393
Homework 2 (Chapter 3,4)
Problems
1. Suppose that A, B and C are three events such that A and B are disjoint, A and C are independent, and B and C are independent. Suppose also that 4P (A) = 2P (B) = P (C) > 0 and
P (A ∪ B ∪ C) = 5P (A). Determine the value of P (A).
2. Suppose 3 coins are tossed. Each coin has an equal probability of head or tail, but are not
independent.
a. What are the minimum and maximum values of the probability of three heads?
b. Now assume that all pairs of coins are mutually independent. What are the minimum and
maximum values of the probability of three heads?
3. In the following figure , the probability of being diconnected from network for each node is
indepent from others and is equal to p. A can be connected to B only when there is a path in
which all the nodes are on.
a. suppose 5 node is disconected , what is the probability of A being connected to B?
b. suppose h , a , d are disconnected and we don’t have any information about the state of other
nodes, what is the probability of connection between A and B?
4. For any three events A, B and D, such that P r(D) > 0, prove that
P (A ∪ B|D) = P (A|D) + P (B|D) − P (A ∩ B|D)
5. In each day stock price goes up or down by one unit with probabilities p and 1 − p respectively,
independent of its behavior on the other days. Given that the stock prices has increased by one
unit after 3 days, what is the probability that the stock went up at the end of first day?
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6. Suppose that an insurance company classifies people into one of three classes; good risks, average
risks, and bad risks. Their record indicates that the probabilities that good, average, and bad
risk persons will be involved in an accident over a one-year span are, respectively, .10, .20 and
.40. If 10 percent of the population are good risks, 50 recent are average risks, and 40 percent
are bad risks, what proportion of people have accidents in a fixed year? If policy-holder A had
no accidents in 2013, what is the probability that he or she is a good risk?
7. Suppose that when a machine is adjusted properly, 50 percent of the items produced by it are
of high quality and the other 50 percent are of medium quality. Suppose, however, that the
machine is improperly adjusted during 10 percent of the time and that, under these conditions,
25 percent of the items produced by it are of high quality and 75 percent are of medium quality.
a. Suppose that five items produced by the machine at a certain time are selected at random
and inspected. If four of these items are of high quality and one item is of medium quality,
what is the probability that the machine was adjusted properly at that time?
b. Suppose that one additional item, which was pro- duced by the machine at the same time
as the other five items, is selected and found to be of medium quality. What is the new
posterior probability that the machine was adjusted properly?
8. In a test for detecting a particular type of genetic disorder ,the probability that a person who
has this type of disorder have a positive reaction to the test is 0.95 and the probability that the
person have a negative reaction is 0.05. If the test is applied to a person who does not have
this type of genetic disorder, the probability of positive reaction is 0.05 and the probability of
negative reaction is 0.95. Suppose that in the general population, one person out of every 10,000
people has this type of genetic disorder. If a person selected at random has a positive reaction
to the test, what is the probability that he has this type of genetic disorder?
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