# Math 308A Spring 2009 Study Sheet, Sample Test Questions

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```					Math 308A Spring 2009: Study Sheet, Sample Test Questions
You will have a sheet of formulae. Your first job: (1) examine the
formulae sheet, (2) let me know if there are other formulae that
should be included, and (3) use the formulae sheet to practice for
the exam. Know the assumptions for all distributions. If you do
not know the assumptions of the distributions, the formulae
sheet is just a tool that you do not know how to use.
Text Section                              Assigned problems
3.1: Sample space and events:              3.13
Experiment, Sample space, Event,
Outcome, Set operations, Venn
diagrams
3.2: Counting: Tree diagrams, Thm         3.15, 18, 19, 20, 21, 22, 24
3.1-Thm3.3, Combinations,
Permutations, Factorials
3.3: Probability: Classical and
frequency probability
3.4: Axioms of Probability:               Know the axioms and how
Probability as an additive set function,  to use them
Axioms of probability
3.5: Some elementary theorems:            3.28, 30, 33, 3.7(d) and
Thm 3.4-Thm 3.7—probability of union      3.37(a), 42, 45, 51, 52
and complement

3.6: Conditional probability: Thm        3.58, 62, 64, 66(b), 68, 70
3.8 and Them 3.9

3.7: Bayes theorem: Thm 3.10, Thm        3.73, 74, 76, 78, 80
3.11
3.8: Mathematical expectation and
decision making: definition of
mathematical expectation of a discrete
r.v. at the top of p. 93—It is the
distribution mean.
4.1: Random Variables: random
variable, probability distribution
function, cumunlative distribtution
function or distribution function
4.2 Binomial Distribution: know all,     4, 9, 12 (f), 18
including the assumptions
4.3 Hypergeometric Distribution          24, 26
4.4 Mean and variances of                32, 36, 43 (43 refers to 4.2)
probability distribution
4.5 Chebyshev’s theorem                                44, 46

Sample Questions
1.  If a random variable has the binomial distribution with n=40 and p=.6, find the
probability
(a)     [8] that the variable will take a value less than 12
(b)     [6] that the variable will take the value 12.

2.     A quality control inspector accepts a shipment whenever a sample of size 5
contains no defectives. Otherwise, the inspector rejects the shipment. Find the
probability that the inspector rejects a shipment of 100 items in which 3 are
defective: (a) [8] for sampling with replacement; (b) [6] for sampling without
replacement.
3.     Three machines produce the total output of a factory. Machine 1 produces
50% of the output. Machine 2 produces 35% and machine 3 produces 15% of the
output. Eight percent of the output of machine 1 is defective, 10% of the output of
machine 2 is defective, and, 7% of the output of machine 3 is defective. (a) [8]
What is the probability that a randomly selected item will be defective?
(b) [7] If a randomly chosen item is defective, what is the probability that it was produced
by machine 3?
4.     Managers assume that fifteen percent of widgets made in one day are
defective. If this assumption is correct, what is the exact probability that more
than 18 of 100 randomly selected widgets will be defective.
c
5.     The probability distribution function of x is   f ( x)       , x  0,1, 2,3,... . Find c.
4x
3. [6] Shade the area corresponding to ( A  C )  B on a Venn diagram.
4. [8] How many permutations are there of the letters in the word “zebra”?
5. [8] How many permutations are there of the letters in “elephant” if we consider
the two “e”s to be identical but all other letters to be distinct?
6. (a) [7] If a circuit board plant has five teams competing for an efficiency prize, in
how many ways can a winning team and a second-place team be selected?
7. A nondestructive test for corrosion of the inside of a section of pipe in the
cooling system of a nuclear plant has probability of 0.75 of detecting
corrosion when it is present and a probability of 0.2 of indicating corrosion
when corrosion is not present. The actual rate of corrosion seems to be about
0.08.
(a) [8] Determine the probability that a section of pipe has internal corrosion
given that the test indicates the presence of corrosion.
(b) [8] Determine the probability that a section of pipe has internal corrosion
given that the test does NOT indicate the presence of corrosion.
8. [8] A factory has three trainers for line personnel. Trainer A trains 40% of all
new hires, trainer B trains 25% and trainer C trains 35%. If 80% of trainer
A’s trainees pass competency test, 55% of trainer B’s trainees pass and 70%
of trainer C’s trainees pass, what is the probability that a randomly selected
trainee passes the competency test?

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