MATH 384. Guía 5.
1. An experiment involves tossing a single die. These are some events:
A: Observe a 2
B: Observe an even number
C: Observe a number greater than 2
D: Observe both A and B
E: Observe A or B or both
F: Observe both A and C
a. List the simple events in the sample space.
b. List the simple events in each of the events A through F.
c. What probabilities should you assign to the simple events?
d. Calculate the probabilities of the six events A through F by adding the appropriate
simple-event probabilities.
2. A sample space S consists of five simple events with these probabilities:
P(E1)=P(E2)=0.15; P(E3)=0.4; P(E4)=2P(E5)
a. Find the probabilities for simple events E4 and E5
b. Find the probabilities for these two events:
A: EI, E3, E4
B: E2, E3
c. List the simple events that are either in event A or event B or both.
d. List the simple events that are in both event A and event B.
3. A sample space contains 10 simple events: EI,E2. . . . , E10. If P(E1) = 3P(E2) =
0.45 and the remaining simple events are equiprobable, find the probabilities of
these remaining simple events.
4. A particular basketball player hits 70% of her free throws. When she tosses a pair
of free throws, the four possible simple events and three of their associated
probabilities are as given in the table:
Simple
Outcome of
Outcome of
Event
First Free Throw
Second Free Throw Probability
_______________________________________________________
1
Hit
Hit
.49
2
Hit
Miss
?
3
Miss
Hit
.21
4
Miss
Miss
.09
a) Find the probability that the player will hit on the forst trowand miss in the
second.
b) Find the probability that the player will hit on at least one of the two free trows.
5. A survey classified a large number of adults according to whether they were
judged to need eyeglasses to correct their reading vision and whether they used
eyeglasses when reading. The proportions falling into the four categories are
shown in the table. (Note that a small proportion, .02, of adults used eyeglasses
when in fact they were judged not to need them.)
Used Eyeglasses for Reading
Judged to Need Eyeglasses
Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from this large group, find the probability of each event:
a. The adult is judged to need eyeglasses.
b. The adult needs eyeglasses for reading but does not use them.
c. The adult uses eyeglasses for reading whether he or she needs them or not.
6. The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are
numbered 1,2, . . .,36, and the remaining two are marked 0 and 00. The wheel is
spun, and a pocket is identified as the "winner." Assume that the observance of
anyone pocket is just as likely as any other.
a. Identify the simple events in a single spin of the roulette wheel.
b. Assign probabilities to the simple events.
c. Let A be the event that you observe either a 0 or a 00. List the simple events in the
event A and find P(A).
d. Suppose you placed bets on the numbers 1 through 18. What is the probability that
one of your numbers is the winner?
7. Three people are randomly selected from voter registration and driving records to
report for jury duty. The gender of each person is noted by the county clerk.
a. Define the experiment.
b. List the simple events in S.
c. If each person is just as likely to be a man as a woman, what probability do you
assign to each simple event?
d. What is the probability that only one of the three is a man?
e. What is the probability that all three are women?
Refer to Exercise 7. Suppose that there are six prospective jurors, four men and two
women, who might be impaneled to sit on the jury in a criminal case. Two jurors are
randomly selected from these six to fill the two remaining jury seats.
a. List the simple events in the experiment (HINT: There are 15 simple events if you
ignore the order of selection of the two jurors.)
b. What is the probability that both impaneled jurors are women?
8. A food company plans to conduct an experiment to compare its brand of tea with
that of two competitors. A single person is hired to taste and rank each of three
brands of tea, which are unmarked except for identifying symbols A, B, and C.
a. Define the experiment.
b. List the simple events in S.
c. If the taster has no ability to distinguish difference in taste among teas, what is the
probability that the taster will rank tea type A as the most desirable? As the least
desirable?
9. Four equally qualified runners, John, Bill, Ed, and Dave, run a l00-meter sprint,
and the order of finish is recorded.
a. How many simple events are in the sample space?
b. If the runners are equally qualified, what probability should you assign to each
simple event?
c. What is the probability that Dave wins the race?
d. What is the probability that Dave wins and John places second?
e. What is the probability that Ed finishes last?
10. In a genetics experiment, the researcher mated two Drosophila fruit flies and
observed the traits of 300 offspring. The results are shown in the table.
Eye Color
Normal
Vermillion
|
|
|
|
Wing Size
Normal Miniature
140
3
6
151
One of these offspring is randomly selected and observed for the two genetic traits.
a. What is the probability that the fly has normal eye color and normal wing size?
a. What is the probability that the fly has vermillion eyes?
a. What is the probability that the fly has either vermillion eyes or miniature wings, or
both?
MATH 384. Guía 6.
11. You have two groups of distinctly different items, 10 in the first group and 8 in
the second. If you select one item from each group, how many different pairs can
you form?
12. You have three groups of distinctly different items, four in the fIrst group, seven
in the second, and three in the third. If you select one item from each group, how
many different triplets can you form?
13. Evaluate the following permutations. (HINT: Your scientific calculator may have
a function that allows you to calculate permutations and combinations quite
easily.)
a. P35
b. P910
c. P66
d. P120
1. Evaluate these combinations:
a. C35
b. C910
c. C66
d. C120
In how many ways can you select fIve people from a group of eight if the order of
selection is important?
In how many ways can you select two people from a group of 20 if the order of selection
is not important?
Three dice are tossed. How many simple events are in the sample space?
Four coins are tossed. How many simple events are in the sample space?
Three balls are selected from a box containing 10 balls. The order of selection is not
important. How many simple events are in the sample space?
You own 4 pairs of jeans, 12 clean T-shirts, and 4 wearable pairs of sneakers. How many
outfits (jeans, T-shirt, and sneakers) can you create?
A businessman in New York is preparing an itinerary for a visit to six major cities. The
distance traveled, and hence the cost of the trip, will depend on the order in which he
plans his route. How many different itineraries (and trip costs) are possible?
Your family vacation involves a cross-country air flight, a rental car, and a hotel stay in
Boston. If you can choose from four major air carriers, five car rental agencies, and three
major hotel chains, how many options are available for your vacation accommodations?
Three students are playing a card game. They decide to choose the fmt person to play by
each selecting a card from the 52-card deck and looking for the highest card in value and
suit. They rank the suits from lowest to highest: clubs, diamonds, hearts, and spades.
a. If the card is replaced in the deck after each student chooses, how many possible
configurations of the three choices are possible?
b. How many configurations are there in which each student picks a different card?
c. What is the probability that all three students pick exactly the same card?
d. What is the probability that all three students pick different cards?
A French restaurant in Riverside, California, offers a special summer menu in which, for
a fixed dinner cost, you can choose from one of two salads, one of two entrees, and one
of two desserts. How many different dinners are available?
Five cards are selected from a 52-card deck for a poker hand.
a. How many simple events are in the sample space?
b. A royal flush is a hand that contains the A, K, Q, J, and 10, all in the same suit. How
many ways are there to get a royal flush?
c. What is the probability of being dealt a royal flush?
Refer to Exercise (five cards). You have a poker hand containing four of a kind.
a. How many possible poker hands can be dealt?
b. In how many ways can you receive four cards of the same face value and one card
from the other 48 available cards?
c. What is the probability of being dealt four of a kind?
A study is to be conducted in a hospital to determine the attitudes of nurses toward
various administrative procedures. If a sample of 10 nurses is to be selected from a total
of 90, how many different samples can be selected? (HINT: Is order important in
determining the makeup of the sample to be selected for the survey?)
Two city council members are to be selected from a total of five to form a subcommittee
to study the city's traffic problems.
a. How many different subcommittees are possible?
b. If all possible council members have an equal chance of being selected, what is the
probability that members Smith and Jones are both selected?
MATH 384. Guía 7.
Identify the following as discrete or continuous random variables:
a. Increase in length of life attained by a cancer patient as a result of surgery
b. Tensile breaking strength (in pounds per square inch) of 1-inch-diameter steel cable
c. Number of deer killed per year in a state wildlife preserve
d. Number of overdue accounts in a department store at a particular time
e. Your blood pressure
A random variable x has this probability distribution:
x
P(x)
0
.1
1
.3
2
.4
3
.1
4
?
5
.05
a. Find p(4).
b. Construct a probability histogram to describe p(x).
c. Find  ,  2 and  .
d. Locate the interval   2 on the x-axis of the histogram. What is the probability that x
will fall into this interval?
e. If you were to select a very large number of values of x from the population, would
most fall into the interval   2 . Explain.
A random variable x can assume five values: 0, I, 2, 3,4. A portion of the probability
distribution is shown here:
x
0
1
2
3
4
P(x)
.1
.3
.3
?
.1
a. Find p(3).
b. Construct a probability histogram for p(x).
c. Calculate the population mean, variance, and standard deviation.
d. What is the probability that x is greater than 2?
e. What is the probability that x is 3 or less?
Let x equal the number observed on the throw of a single balanced die.
a. Find and graph the probability distribution for x.
b. What is the average or expected value of x?
c. What is the standard deviation of x?
d. Locate the interval   2 on the x-axis of the graph in part a. What proportion of all
the measurements would fall into this range?
Let x represent the number of times a customer visits a grocery store in a 1-week period.
Assume this is the probability distribution of x:
x
0
1
2
3
P(x)
.1
.4
.4
.1
Find the expected value of x, the average number of times a customer visits the store.
Who is the king of late night TV? An Internet survey estimates that, when given a choice
between David Letterman and Jay Leno, 52% of the population prefers to watch Jay
Leno. Suppose that