NAME: MAT 226: Review Test III 1. A small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible? 2. A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors, and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen. How many different subcommittees are possible? 3. How many different batting orders are possible for a baseball team consisting of 9 players? 4. A class in probability theory consists of 6 men and 4 women. An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score. (a) How many different rankings are possible? (b) If the men are ranked just among themselves and the women just among themselves, how many different rankings are possible? 5. A committee of 3 is to be formed from a group of 20 people. How many different committees are possible? 6. From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? What if 2 of the men are feuding and refuse to serve on the committee together? 7. Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible? 8. For a trip you pack 5 shirts and 3 pairs of pants, and two jackets. Assuming everything can be worn with everything else, how many dif- ferent combinations can you wear? 9. Suppose you have 6 pictures, and you want to arrange 4 of them along a wall. In how many different ways can you arrange them? 10. A basketball squad has 12 players. (a) If all players can play any position, how many ways can a team of five be chosen? (b) If only four players out of the 12 are able to be the center, how many ways can a team be chosen? 11. A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. (a) Describe the sample space. (b) Describe the sample space, when the second marble is drawn with- out replacing the first marble. 12. A game consisting of flipping a coin ends when the player gets two heads in a row, two tails in a row, or flips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end? 13. Two dice are thrown. Let E be the event that the sum of the dice is odd, let F be the event that at least one of the dice lands on 1, and let G be the event that the sum is 5. Describe the events (a) E ∩ F (b) E ∪ F (c) F ∩ G (.

1. NAME:
MAT 226: Review Test III
1. A small community consists of 10 women, each of whom has
3 children.
If one woman and one of her children are to be chosen as
mother and child of the year, how many different choices are
possible?
2. A college planning committee consists of 3 freshmen, 4
sophomores, 5 juniors, and 2 seniors. A subcommittee of 4,
consisting of 1 person from each class, is to be chosen. How
many different subcommittees are possible?
3. How many different batting orders are possible for a
baseball team consisting of 9 players?
4. A class in probability theory consists of 6 men and 4
women.
An examination is given, and the students are ranked
according to their performance. Assume that no two students
obtain the same score.
(a) How many different rankings are possible?
(b) If the men are ranked just among themselves and the women
just among themselves, how many different rankings are
possible?
5. A committee of 3 is to be formed from a group of 20 people.
How many different committees are possible?
6. From a group of 5 women and 7 men, how many different
committees consisting of 2 women and 3 men can be formed?
What if 2 of the men are feuding and refuse to serve on the
committee together?
7. Ms. Jones has 10 books that she is going to put on her
bookshelf. Of these, 4 are mathematics books, 3 are chemistry

2. books, 2 are history books, and 1 is a language book. Ms. Jones
wants to arrange her books so that all the books dealing with
the same subject are together on the shelf. How many different
arrangements are possible?
8. For a trip you pack 5 shirts and 3 pairs of pants, and
two jackets.
Assuming everything can be worn with everything else, how
many dif- ferent combinations can you wear?
9. Suppose you have 6 pictures, and you want to arrange 4 of
them along a wall. In how many different ways can you arrange
them?
10. A basketball squad has 12 players.
(a) If all players can play any position, how many ways can a
team of five be chosen?
(b) If only four players out of the 12 are able to be the center,
how many ways can a team be chosen?
11. A box contains 3 marbles: 1 red, 1 green, and 1 blue.
Consider an experiment that consists of taking 1 marble
from the box and then replacing it in the box and drawing a
second marble from the box.
(a) Describe the sample space.
(b) Describe the sample space, when the second marble is
drawn with- out replacing the first marble.
12. A game consisting of flipping a coin ends when the player
gets two heads in a row, two tails in a row, or flips the coin four
times.
(a) Draw a tree diagram to show the ways in which the game
can end. (b) In how many ways can the game end?
13. Two dice are thrown. Let E be the event that the sum of
the dice is odd, let F be the event that at least one of the dice
lands on 1,
and let G be the event that the sum is 5. Describe the events
(a) E ∩ F (b) E ∪ F (c) F ∩ G (d) E ∩ F 0
(e) E ∩ F ∩ G

3. 14. Three coins are tossed.
(a) List the elements in the sample space.
(b) Find the probability that exactly two heads show.
15. You flip an unfair coin,
where p(heads) = 3
and p(tails) = 1 , ten times.
(a) Find p(exactly 9 heads). (b) Find p(exactly 7 heads). (c)
Find p(at least 7 heads).
16. A total of 500 married working couples were polled about
their annual salaries, with the following information resulting:
Wife
Husbend
For instance, in 36 of the couples, the wife earned more and the
husband earned less than $25, 000. If one of the couples is
randomly chosen, what is
(a) the probability that the husband earns less than $25, 000?
(b) the conditional probability that the wife earns more than
$25, 000 given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than
$25, 000 given that the husband earns less than this amount?
17. If two fair dice are rolled, what is the conditional
probability that the first one lands on 6 given that the sum of
the dice is 9?
18. What is the probability that a fair coin lands Heads 4
times out of 5 flips?
19. At a party, 1 of the guest are women. 75% percent of the
women wore sandals and 40% of the men wore sandals.
(a) What is the probability that a person chosen at random

4. at the party is a man wearing sandals?
(b) What is the probability that a person chosen at random is
wearing sandals?
20. A manufacturer of front lights for automobiles tests lamps
under a high- humidity, high-temperature environment using
intensity and useful life as the responses of interest. The
following table shows the performance of 140 lamps:
u seful life
satisfactory
unsatisfactory
s atisfactory intensity
76
14
u nsatisfactory intensity
32
18
(a) Find the probability that a randomly selected lamp will
yield sat- isfactory useful life and satisfactory intensity.
(b) Find the probability that a randomly selected lamp has
satisfac- tory useful life given that it has satisfactory intensity.
21. An exam consists of ten true-or-false questions. If a student
guesses at every answer, what is the probability that he or she
will answer exactly six questions correctly?
22. A survey is done of people making purchases at a gas
station
buy drink
no drink
Total
buy gas
20
15
35
no gas
10

5. 5
15
Total
30
20
50
(a) What is the probability that a person buys a drink?
(b) What is the probability that a person doesnt buy a drink?
(c) What is the probability that a person buys gas and a
drink?
(d) What is the probability that a person buys gas but not a
drink?
(e) What is the probability that a person who buys a drink
also buys gas?
(f ) What is the probability that a person who doesnt buy a
drink buys gas?
23. Two fair dice are thrown. Let E denote the event that the
sum of the dice is 7. Let F denote the event that the first die
equals 4. Are E and F are independent events?
24. Celine is undecided as to whether to take a French course
or a chem- istry course. She estimates that her probability of
receiving an A grade
would be 1
in a French course and 2
in a chemistry course. If Ce-
line decides to base her decision on the flip of a fair coin, what
is the probability that she gets an A in chemistry?
25. A medical experiment showed the probability that a new
medicine was effective was .75, the probability of a certain
side effect was .4, and the probability of both occurring was .3.
Are the events independent?
26. In Orange County, 51% of the adults are males. (It doesn’t
take too much advanced mathematics to deduce that the other
49% are females.) One adult is randomly selected for a survey

6. involving credit card usage.
(a) Find the prior probability that the selected person is a
male.
(b) It is later learned that the selected survey subject was
smoking a cigar. Also, 9.5% of males smoke cigars, whereas
1.7% of fe- males smoke cigars. Use this additional
information to find the probability that the selected subject is
a male.
27. An insurance company believes that people can be divided
into two classes: those who are accident prone and those
who are not. The companys statistics show that an accident-
prone person will have an accident at some time within a
fixed 1-year period with probability
0.4, whereas this probability decreases to 0.2 for a person who
is not accident prone. If we assume that 30% of the
population is accident prone.
(a) What is the probability that a new policyholder will have
an ac- cident within a year of purchasing a policy?
(b) Suppose that a new policyholder has an accident within a
year of purchasing a policy. What is the probability that he
or she is accident prone?
28. (6 pt) A manufacturer claims that its drug test will detect
steroid use (that is, show positive for an athlete who uses
steroids) 95% of the time. What the company does not tell
you is that 15% of all steroid- free individuals also test
positive (the false positive rate). 10% of the rugby team
members use steroids.
Your friend on the rugby team has just tested positive.
What is the probability that your friend use steroid?
29. (5 pt each) X is the outcome when we roll a pair fair dice.
(a) Find E[X ] the expected value.
(b) Find V (X ) the variance of X .
30. A pair of dice, each with the numbers 1,2,2,3,3,3 on its six
sides are rolled.
(a) What is the expected value of the sum of the numbers
showing? (b) What is the V (X ) the variance ?

7. 31. A contestant on a quiz show is presented with two
questions, questions
1 and 2, which he is to attempt to answer in some order he
chooses. If he decides to try question i first, then he will be
allowed to go on to question j, j = i, only if his answer to
question i is correct. If his initial answer is incorrect, he is not
allowed to answer the other question. if he is 60% certain of
answering question 1, worth$200, correctly and he is 80%
certain of answering question 2, worth $100, correctly. Which
question he should try first as to maximize his expected
winnings?
32. Let X denote a random variable that takes on any of the
values ?1, 0, and 1 with respective probabilities
P (X = −1) = 0.2, P (X = 0) = 0.5, P (X = 1) = 0.3
Compute E(X 2 )
33. In a gambling game, a woman is paid $3 if she draws a jack
or a queen and $5 if she draws a king or an ace from an ordinary
deck of 52 playing cards. If she draws any other card, she
loses. How much should she pay to play if the game is fair?
34. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1,
respectively, that 0, 1, 2, or 3 power failures will strike a
certain subdivision in any given year.
Find the mean and variance of the random variable X
representing the number of power failures striking this
subdivision.
35. The probability that a certain kind of component will
survive a shock test is 3/4. Find the probability that exactly 2
of the next 4 components tested survive.
The probability that a patient recovers from a rare blood
disease is
0.4. If 15 people are known to have contracted this disease,
what is the probability that
(a) at least 10 survive? (b) from 3 to 8 survive? (c) exactly 5
survive?

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