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Math 309
Test 2
Show all work in order to receive credit.
Carter
Name__________________________
10/6/2000
1. Suppose that each child born to a couple is equally likely to be a boy or a girl independent of the sex distribution of the
other children in the family. For a couple having 5 children, compute the probabilities that:
a) all children are of the same sex.
b) the 3 eldest are boys and the others girls.
c) exactly three are boys.
d) there is at least one girl.
2. Forty percent of a particular population has type A blood. A group randomly selected from the population has their
blood typed. Find the probability that the first person with type A blood is the fifth person tested.
3. You pay a $60 annual premium on an insurance policy for a $500 car stereo system. The insurance company has
determined the probability of loss in any one year to be 0.1. Let G= the insurance company’s gain from sale of the policy.
a) Give the probability mass function for G.
b) Find the expected gain.
4. A group consists of 10 senior engineering majors, 5 junior engineering majors, 8 senior sciences majors, and 12 junior
sciences majors. A person is randomly selected from the group. Let E = the event that the person chosen is a senior and F=
the event that the person chosen is a sciences major.
a) Are E and F mutually exclusive? Justify your response.
b) Are E and F independent? Justify your response.
5. An urn contains 3 blue balls, 4 red balls, and 1 green ball. You select three balls without replacement. Let X = the
number of blue balls chosen. Find the probability that X is at least two. Give the probability mass function for X.
6. Find the cumulative distribution function for X. Sketch its graph.
X
0
2
5
P(X)
.20
.45
.35
7. A communication network has a safeguard system. If line 1 fails, it is bypassed and line 2 is used. If line 2 also fails, it is bypassed
and line 3 is used. The probability that any one line fails is 0.01 and failures are independent of each other.
a) What is the probability that a communication gets through the system?
x 1
0
b) If a communication gets through, what is the probability that it used line 2?
8. Given the cumulative distribution function to the right. P(2 X  4.5) = _______
9.
.3
1  x3

F ( x)  .5
3  x  4.5
.85 4.5  x  10

 1
x  10
An appliance store sells two brands of VCRs. 70% of the sales are brand 1. If a customer purchases brand 1, the
probability that he/she purchases an extended warranty is 20%, while 40% of those purchasing brand 2 buy an
extended warranty. Find the probability that a customer who purchases a VCR also purchases an extended warranty.
10. Select the most appropriate answer.
A random variable is:
a) a variable in an equation whose value is random.
b) a probability mass function.
c) a function which assigns numbers to a sample space.
d) an outcome of an experiment.
11. Using the axioms of probability, derive P(AB) = P(A) + P(B) – P(AB).
12. Prove that if E and F are independent, then Ec and F are independent.
__________________________________________________________________________________________________
Ans. 1) 0.55+ 0.55, 0.53*0.52, C(5,3)* 0.53*0.52, 1-0.55; 2) 0.64*0.4 3) p(60) = .9 p(-440) = .1; E[G] = 10;
(4) a. No, there are seniors who major in the sciences; P(senior)=18/35, P(senior | sciences) = 8/20since these probabilities
are not equal, the events are independent; 5) P(X>=2) = (C(3,2)*C(5,1)+C(3,3))/C(8,3); P(X) = C(3,x)*C(5, 3-x)/C(8,3)
x=0,1,2,3; 6) P(X<0) = 0, P(0<=X<2) = 0.2, P(2<=X<5) = 0.65, P(X >= 5) =1; 7) 1-(.01)3, 100/10101; 8) .85;
9) (.7)(.2)+(.3)(.4)
Math 309
Test 2
Show all work in order to receive credit.
Carter
Name__________________________
10/6/2000
1. A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80%
chance that she will get the job if she receives a strong recommendation, a 40% chance is she receives a moderately good
recommendation, and a 10% chance if she receives a weak recommendation. She also estimates that the probabilities that
the recommendation will be strong, moderate, or weak are .7, .2, and .1 respectively.
a) Under these assumptions, what is the probability that she receives the new job offer?
b) If she receives the job offer, what is the probability that she received a strong recommendation?
2. Consider a system of water flowing through valves from A to B. Valves 1, 2, and 3 operate independently. Each
correctly opens on signal with probability .8. Let X = the number of open paths from A to B after the signal is given. Give
the probability mass function for X.
3. A fire detection device contains three temperature-sensitive cells. Each acts independently of the others and any one
may activate an alarm. Each has probability .92 of activating the alarm when the temperature reaches the danger level.
a) If the temperature reaches the danger level, what is the probability that the alarm sounds?
b) If the temperature reaches the danger level, what is the probability that the alarm doesn’t sound?
c) What is the probability that exactly two of the three cells activate the alarm?
4. A pair of dice is rolled until both dice simultaneously show six. What is the probability that you roll the pair of dice ten
times? Let Y = the number of the roll when the pair of sixes appears. Give the probability mass function for Y.
5a) Sketch the cumulative distribution function defined in the box.
b) Give the associated probability mass function.
x 1
0
.3
1  x3

F ( x)  .5
3  x  4.5
.85 4.5  x  10

 1
x  10
6. A spinner is numbered 0, 1, 2, 3, 4, 5. Each number is equally likely to occur. You bet $1 on one number. If the
pointer rests on that number after the spin, you win $4 (and keep your $1 bet). If the pointer rests on any of the other
numbers, you lose your bet.
a) Give the probability mass function for your gain when playing the game.
b) What is your expected profit? Interpret your answer.
7. A box contains five red pens, two blue pens, and six black pens. You randomly grab three pens from the box. Let X=
the number of blue pens chosen. Give the probability mass function for X in table form.
8. According to a mortality table, the probability that a 35-year-old U.S. citizen will live to age 65 is 0.725. Stan and
Oliver are unrelated, 35-year-old Americans. What is the probability that at least one of them lives to age 65?
9. Select the most appropriate answer.
A random variable is:
a) a variable in an equation whose value is random.
b) a probability mass function.
c) a function which assigns numbers to a sample space.
d) an outcome of an experiment.
13. A group consists of juniors and seniors whose majors are in sciences or engineering. There are 10 senior engineering
majors, 4 senior sciences majors, and 6 junior sciences majors. Let A= {being a junior} and B = {being an engineering
major}
a) How many junior engineering majors are there if the events A and B are independent?
b) How many junior engineering majors are there if the events A and B are mutually exclusive?
11. Prove that if E and F are independent, then E c and Fc are independent.
12. Show that P(AB)  P(A) + P(B) - 1.