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PROBABILITY TOPICS: HOMEWORK
Susan Dean and Barbara Illowsky (2012)
EXERCISE 1
Suppose that you have 8 cards. 5 are green and 3 are yellow. The 5 green cards are numbered
1, 2, 3, 4, and 5. The 3 yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You
randomly draw one card. Consider the following events:
G = card drawn is green; E = card drawn is even-‐numbered
a. List the sample space.
b. P(G) =
c. P(G|E) =
d. P(G and E) =
e. P(G or E) =
f. Are G and E mutually exclusive? Justify your answer numerically.
EXERCISE 2
Refer to problem (1) above. Suppose that this time you randomly draw two cards, one at a
time, and with replacement.
G
= first card is green; G 2
= second card is green.
a. Draw a tree diagram of the situation.
b. P( G 1
and G 2
c. P(at least one green) =
d. P(G 2
|G
e. Are G 2
and G 1
independent events? Explain why or why not.
EXERCISE 3
Refer to problem (1) above. Suppose that this time you randomly draw two cards, one at a
time, and without replacement.
G
= first card is green; G 2
= second card is green.
a. Draw a tree diagram of the situation.
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b. P( G 1
and G 2
c. P(at least one green) =
d. P(G 2
|G
e. Are G 2
and G 1
independent events? Explain why or why not.
EXERCISE 4
Roll two fair dice. Each die has 6 faces.
a. List the sample space.
b. Let A be the event that either a 3 or 4 is rolled first, followed by an even number. Find
P(A).
c. Let B be the event that the sum of the two rolls is at most 7. Find P(B).
d. In words, explain what “P(A|B)” represents. Find P(A|B).
e. Are A and B mutually exclusive events? Explain your answer in 1 -‐ 3 complete sentences,
including numerical justification.
f. Are A and B independent events? Explain your answer in 1 -‐ 3 complete sentences,
including numerical justification.
EXERCISE 5
A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a
card is picked, the color of it is recorded. An experiment consists of first picking a card and then
tossing a coin.
a. List the sample space.
b. Let A be the event that a blue card is picked first, followed by landing a head on the coin
toss. Find P(A).
c. Let B be the event that a red or green is picked, followed by landing a head on the coin
toss. Are the events A and B mutually exclusive? Explain your answer in 1 -‐ 3 complete
sentences, including numerical justification.
d. Let C be the event that a red or blue is picked, followed by landing a head on the coin
toss. Are the events A and C mutually exclusive? Explain your answer in 1 -‐ 3 complete
sentences, including numerical justification.
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J and K are independent events. P(J|K) = 0.3. P(K) = 0.5. Find P(J).
EXERCISE 11
U and V are mutually exclusive events. P(U) = 0.26; P(V) = 0.37. Find:
a. P(U and V)
b. P(U | V)
c. P(U or V)
EXERCISE 12
Q and R are independent events. P(Q) = 0.4; P(Q and R) = 0.10. Find P(R).
EXERCISE 13
Y and Z are independent events.
a. Rewrite the basic Addition Rule ( P(Y or Z) = P(Y) + P(Z) -‐ P(Y and Z) ) using the
information that Y and Z are independent events.
b. Use the rewritten rule to find P(Z) if P(Y or Z) = 0.71 and P(Y) = 0..
EXERCISE 14
G and H are mutually exclusive events. P(G) = 0.5; P(H) = 0.3.
a. Explain why the following statement MUST be false: P(H|G) = 0..
b. Find: P(H or G).
c. Are G and H independent or dependent events? Explain in a complete sentence.
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EXERCISE 15
The following are real data from Santa Clara County, CA. As of March 31, 2000, there was a total
of 3059 documented cases of AIDS in the county. They were grouped into the following
categories (Source: Santa Clara County Public H.D.):
Risk Factors
Gender Homosexual/
Bisexual
IV Drug
**User ***
Heterosexual
Contact
Other
female 0 70 136 49
male 2146 463 60 135
- includes homosexual/bisexual IV drug users
Suppose one of the persons with AIDS in Santa Clara County is randomly selected. Compute the
following:
a. P(person is female) = _____
b. P(person has a risk factor heterosexual contact) = _____
c. P(person is female OR has a risk factor of IV Drug User) = _____
d. P(person is female AND has a risk factor of homosexual/bisexual) = _____
e. P(person is male AND has a risk factor of IV Drug User) = _____
f. P(female GIVEN person got the disease from heterosexual contact) = _____
g. Construct a Venn Diagram. Make one group females and the other group
heterosexual contact. Fill in all values as integers.
EXERCISE 16
Solve these questions using probability rules. Do NOT use the contingency table above. 3059
cases of AIDS had been reported in Santa Clara County, CA, through March 31, 2000. Those
cases will be our population. Of those cases, 6.4% obtained the disease through heterosexual
contact and 7.4% are female. Out of the females with the disease, 53.3% got the disease from
heterosexual contact.
a. P(person is female) = _____
b. P(person obtained the disease through heterosexual contact) = _____
c. P(female GIVEN person got the disease from heterosexual contact) = _____
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For (a) -‐ (e), suppose that you randomly select one player from the 49ers or Cowboys.
a. Find the probability that his shirt number is from 1 to 33.
b. Find the probability that he weighs at most 210 pounds.
c. Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.
d. Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210
pounds.
e. Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at
most 210 pounds.
f. If having a shirt number from 1 to 33 and weighing at most 210 pounds were
independent events, then what should be true about P(shirt # 1 -‐ 33 | ≤ 210
pounds)?
EXERCISE 19
Approximately 249,000,000 people live in the United States. Of these people, 31,800,000 speak
a language other than English at home. Of those who speak another language at home, over 50
percent speak Spanish. (Source: U.S. Bureau of the Census, 1990 Census)
Let: E = speak English at home; E' = speak another language at home; S = speak Spanish
at home
Finish each probability statement by matching the correct answer.
a. P(E' ) = i. 0.
b. P(E) = ii. > 0.
c. P(S) = iii. 0.
d. P(S | E' ) = iv. > 0.
EXERCISE 20
The probability that a male develops some form of cancer in his lifetime is 0.4567 (Source:
American Cancer Society). The probability that a male has at least one false positive test result
(meaning the test comes back for cancer when the man does not have it) is 0.51 (Source: USA
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Today ). Some of the questions below do not have enough information for you to answer them.
Write “not enough information” for those answers.
Let: C = a man develops cancer in his lifetime; P = man has at least one false positive
a. Construct a tree diagram of the situation.
b. P(C) = __________
c. P(P|C) = __________
d. d. P(P|C' ) = __________
e. If a test comes up positive, based upon numerical values, can you assume that man
has cancer? __________ Justify numerically and explain why or why not.
EXERCISE 21
In 1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-‐
citizens to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately
6.5 million people who entered this lottery. Let G = won Green Card
a. What was Renate’s chance of winning a Green Card? Write your answer as a
probability statement.
b. In the summer of 1994, Renate received a letter stating she was one of 110,
finalists chosen. Once the finalists were chosen, assuming that each finalist had an
equal chance to win, what was Renate’s chance of winning a Green Card? Let F =
was a finalist. Write your answer as a conditional probability statement.
c. Are G and F independent or dependent events? Justify your answer numerically and
also explain why.
d. Are G and F mutually exclusive events? Justify your answer numerically and also
explain why.
P.S. Amazingly, on 2/1/95, Renate learned that she would receive her Green Card -‐-‐ true
story!
EXERCISE 22
Three professors at George Washington University did an experiment to determine if
economists are more selfish than other people. They dropped 64 stamped, addressed
envelopes with $10 cash in different classrooms on the George Washington campus.
44% were returned overall. From the economics classes 56% of the envelopes were
returned. From the business, psychology, and history classes 31% were returned.
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h. Comparing “Race and Sex” to “Age,” which two groups are mutually exclusive? How do
you know?
i. Are being male and committing suicide over age 64 independent events? How do you
know?
The following refers to questions (24) and (25): The percent of licensed U.S. drivers (from a
recent year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are
age 20 -‐ 64; 13.61% are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and
under; 81.43% are age 20 -‐ 64; 13.53% are age 65 or over. (Source: Federal Highway
Administration, U.S. Dept. of Transportation)
EXERCISE 24
Complete the following:
a. Construct a table or a tree diagram of the situation.
b. P(driver is female) = __________
c. P(driver is age 65 or over | driver is female) = __________
d. P(driver is age 65 or over AND female) = __________
e. In words, explain the difference between the probabilities in part (c) and part (d).
f. P(driver is age 65 or over) = __________
g. Are being age 65 or over and being female independent events? How do you know?
h. Are being age 65 or over and being female mutually exclusive events? How do you
know?
EXERCISE 25
Suppose that 10,000 U.S. licensed drivers are randomly selected.
a. How many would you expect to be male?
b. Using the table or tree diagram from problem (21), construct a contingency table of
gender versus age group.
c. Using the contingency table, find the probability that out of the age 20 -‐ 64 group, a
randomly selected driver is female.
EXERCISE 26
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Approximately 86.5% of Americans commute to work by car, truck or van. Out of that group,
84.6% drive alone and 15.4% drive in a carpool. Approximately 3.9% walk to work and
approximately 5.3% take public transportation. (Source: Bureau of the Census, U.S. Dept. of
Commerce. Disregard rounding approximations.)
a. Construct a table or a tree diagram of the situation. Include a branch for all other
modes of transportation to work.
b. Assuming that the walkers walk alone, what percent of all commuters travel alone to
work?
c. Suppose that 1000 workers are randomly selected. How many would you expect to
travel alone to work?
d. Suppose that 1000 workers are randomly selected. How many would you expect to
drive in a carpool?
EXERCISE 27
Explain what is wrong with the following statements. Use complete sentences.
a. If there’s a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then
there’s a 130% chance of rain over the weekend.
b. The probability that a baseball player hits a home run is greater than the probability that
he gets a successful hit.
Try these multiple choice questions.
Questions 28 – 29 refer to the following probability tree diagram which shows tossing an unfair
coin FOLLOWED BY drawing one bead from a cup containing 3 red (R), 4 yellow (Y) and 5 blue (B)
beads. For the coin, P(H) = 2/3 and P(T) = 1/3 where H = “heads” and T = “tails.”
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D.
Questions 30 – 32 refer to the following table of data obtained from www.baseball-‐
almanac.com showing hit information for 4 well known baseball players.
Type of Hit
NAME Single Double Triple Home Run TOTAL HITS
Babe Ruth 1517 506 136 714 2873
Jackie Robinson 1054 273 54 137 1518
Ty Cobb 3603 174 295 114 4189
Hank Aaron 2294 624 98 755 3771
TOTAL 8471 1577 583 1720 12351
EXERCISE 30
Find P(hit was made by Babe Ruth)
A.
B.
C.
D.
EXERCISE 31
Find P(hit was made by Ty Cobb | the hit was a Home Run)
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A.
B.
C.
D.
EXERCISE 32
Are the hit being made by Hank Aaron and the hit being a double independent?
A. Yes, because P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)
B. No, because P(hit by Hank Aaron | hit is a double) ≠ P(hit is a double)
C. No, because P(hit by Hank Aaron | hit is a double) ≠ P(hit by Hank Aaron)
D. Yes, because P(hit by Hank Aaron | hit is a double) = P(hit is a double)
EXERCISE 33
Given events G and H: P(G) = 0.43 ; P(H) = 0.26 ; P(H and G) = 0.
A. Find P(H or G)
B. Find the probability of the complement of event (H and G)
C. Find the probability of the complement of event (H or G)
EXERCISE 34
Given events J and K: P(J) = 0.18 ; P(K) = 0.37 ; P(J or K) = 0.
A. Find P(J and K)
B. Find the probability of the complement of event (J and K)
C. Find the probability of the complement of event (J or K)
EXERCISE 35
United Blood Services is a blood bank that serves more than 500 hospitals in 18 states.
According to their website, http://www.unitedbloodservices.org/humanbloodtypes.html, a
person with type O blood and a negative Rh factor (Rh−) can donate blood to any person
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At a certain store the manager has determined that 30% of customers pay cash and 70%
of customers pay by debit card. (No other method of payment is accepted.)
Let M = event that a customer pays cash and D= event that a customer pays by debit card.
Suppose two customers (Al and Betty) come to the store. Explain why it would be
reasonable to assume that their choices of payment methods are independent of each
other.
A. Draw the tree that represents the all possibilities for the 2 customers and their
methods of payment. Write the probabilities along each branch of the tree.
B. For each complete path through the tree, write the event it represents and find the
probability.
C. Let S be the event that both customers use the same method of payment. Find
P(S).
D. Let T be the event that both customers use different methods of payment. Find
P(T) by two different methods: by using the complement rule and by using the
branches of the tree. Your answers should be the same with both methods.
E. Let U be the event that the second customer uses a debit card. Find P(U).
EXERCISE 40
A box of cookies contains 3 chocolate and 7 butter cookies. Miguel randomly selects a
cookie and eats it. Then he randomly selects another cookie and eats it also. (How many
cookies did he take?)
A. Are the probabilities for the flavor of the SECOND cookie that Miguel selects
independent of his first selection, or do the probabilities depend on the type of
cookie that Miguel selected first? Explain.
B. Draw the tree that represents the possibilities for the cookie selections. Write the
probabilities along each branch of the tree.
C. For each complete path through the tree, write the event it represents and find the
probabilities.
D. Let S be the event that both cookies selected were the same flavor. Find P(S).
E. Let T be the event that both cookies selected were different flavors. Find P(T) by
two different methods: by using the complement rule and by using the branches of
the tree. Your answers should be the same with both methods.
F. Let U be the event that the second cookie selected is a butter cookie. Find P(U).
EXERCISE 41
When the Euro coin was introduced in 2002, two math professors had their statistics
students test whether the Belgian 1 Euro coin was a fair coin. They spun the coin rather
than tossing it, and it was found that out of 250 spins, 140 showed a head (event H) while
110 showed a tail (event T). Therefore, they claim that this is not a fair coin.
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A. Based on the data above, find P(H) and P(T).
B. Use a tree to find the probabilities of each possible outcome for the experiment
of tossing the coin twice.
C. Use the tree to find the probability of obtaining exactly one head in two tosses of
the coin.
D. Use the tree to find the probability of obtaining at least one head.
