Probability Homework
Section P5
1. A pair of fair dice are tossed. What is the conditional probability that the two dice are the same given
that the sum equals 8?
2. A die is tossed. a) Find the probability that a prime number was obtained, given that an even number
came up. b) Find the probability that an even number came up given that a prime number came up.
3. Three cards are drawn from a deck of 52 cards without replacement. Find the conditional probability
that the third card is a spade given that the first two cards are spades.
4. A family has three children. Assume that each child is as likely to be a boy as it is a girl. Find the
probability that the family has three girls if it is known that the family has at least one girl.
5. Suppose that the probability that a child of a particular set of parents will have blue eyes is ¼. If the
couple plans to have three children, what is the probability that all three will have blue eyes?
6. Suppose 68% of the citizens of a state approve of the governor, but only 47% approve of both the
governor and the lieutenant governor. If a randomly selected citizen approves of the governor, what is
the probability that the citizen also approves of the lieutenant governor?
7. A card is randomly drawn from a deck of 52 playing cards. Find the probability that it is a red king
given that the card is a face card. (Jacks, queens and kings are face cards.)
8. A survey claims that 70% of the households in a certain town have a color TV, 20 % have a
microwave oven, and 2% have both a color TV and a microwave oven. Find the probability that a
randomly selected household will have a microwave oven, given that it has a color TV.
9. Two cards are drawn in succession from a deck of 52 cards. Find the probability that both cards are
black if the first card is replaced before the second is drawn.
10. A bag contains five red, and twelve white marbles. Two marbles are drawn in succession without
replacement. a) Find the probability that at least one red marble is drawn. b) Find the probability if the
marble is replaced after the first drawing.
11. Given a family has two children (Assume boys and girls are equally likely),
a) If it is known that at least one of the children is a girl, find the probability that both are girls.
b) If the older child is a girl, find the probability that both children are girls.
12. A fair die is rolled 5 times. What is the probability of getting exactly two 5’s?
13.A hand of five cards are drawn from a deck of 52 cards. What is the probability that at least one king
is drawn?
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14. Three cards are drawn at random form a standard deck of 52 cards (without replacement). What is
the probability that the cards are 2 Aces and a 9? (There are 4 aces and 4 9’s in a deck)
15. A random sample of 400 adults are classified according to sex and highest education level completed
as shown in the table.
Education
Female
Male
Elementary school
64
53
High school
99
87
College
28
39
Graduate school
14
16
If a person is picked at random from this group, find the probability that
a) the person is female given that the person has a graduate degree.
b) the person is male given that the person has a high school diploma as his or her highest level of
education completed.
c) the person does not have a college degree given that the person is female.
16. According to the U.S. Center for Health Statistics, in the United States in 1993, 36.3% of hospital
patients were over 64 years old. Whereas on 37.8% of all hospital patients used Medicare as the
principal means of payment, 90.3% of patients over 64 years old did. What is the probability that a
randomly selected patient was over 64 years old and used Medicare as the principal means of payment?
17. The U.S. Bureau of Labor Statistics reports that of those unemployed in 1994, 57.1% were male,
13.4% were college graduates, and 7.6% were male college graduates. What is the probability that a
random unemployed male in 1994 was a college graduate?
18. A survey of college freshmen conducted by the American Council on Education and UCLA in 1996
found 31.7% of women and 5.7% of men plan to become elementary or secondary school teachers. In
1996, women made up approximately 56% of the total freshmen. What is the probability that a
randomly selected college freshman was a woman planning to become an elementary school teacher?
19. Suppose a company produces machine parts and 2% of the parts produced are defective. A test that
the parts are put through detects 92% of the defective parts. What is the probability that a randomly
selected part will be found to be defective?
20. According to the US Bureau of Census estimates, 46,995,000 people in the United States were not
covered by health insurance in 2006. Of these, 13,933,000 had household incomes less than $25,000 per
year. The percentage of the total U.S. population of 296,450,000 in 2006 with household incomes less
than $25,000 per year was 25.2%. What is the probability that in 2006 a random person with a
household income less than $25,000 was not covered by health insurance?
21. Assume that the air quality in a given town is unhealthy on 1 out of 4 days. What is the probability
that the air quality will be unhealthy on exactly 4 out of 10 days?
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22. Suppose that an oil tanker breaks up off the coast of New England, leaving a tremendous oil slick on
the ocean. Suppose further that this oil spill will result in a very bad ecological (and economic) disaster,
unless the oil is blown by a west wind out to the deeper waters before it sinks. If the probability of a
west wind on any day is 1/3, find the probability of getting more than 4 days of west wind in one week.
23. It is known that a certain prescription drug produces undesirable side effects in 30% of all patients
who use it. Among a random sample of eight patients using the drug, find the probability of
a) None have undesirable effects.
b) Exactly two have undesirable effects
c) More than two have undesirable effects
24. A certain brand of tomato seeds has a 75% probability of germinating. To increase the chance that
at least one tomato plant per seed hill germinates, a gardener plants four seed in each hill.
What is the probability that at least one seed germinates in a given hill?
25. An extensive survey revealed that, during a certain presidential election campaign, 64% of the
political columns in a certain group of major newspapers were favorable to the incumbent president. If a
sample of 15 of these columns is selected at random, what is the probability that exactly ten of them will
be favorable?
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Answers
1. F: sum equals 8; E: two dice are the same. n(F) = 5; n(E∩F) = 1; P(E/F) = 1/5
2. a)
b)
. E: 3rd is spade: P(E∩F) =
3. F: 1st two cards are spades; P(F) =
So P(E/F) =
. You can also get this by just asking once two spades are gone (the given),
what is the probability of getting a spade from the cards that are left?
4. F: the family has at least one girl; n(F) = 7 ways. E: family has three girls. n(E∩F) = 1, So P(E/F) =
1/7
5.These are independent events, so P(all 3 have blue eyes) =
=
.
6. F: approves of governor; P(F) = .68; E∩F: approve of both; P(E∩F) = .47, so P(E/F) =
7. F: face cards: n(F) = 12; E: red kings. n(E∩F) = 2; So P(E/F) = 2/12 = 1/6
8. P(Micro/TV) =
P( Micro TV )
P(TV )
0.02
0.7
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9. Independent events so just multiply the probabilities. P(both black) =
.
10.a) (the red can be on the 1st draw or the 2nd draw:2 ways or they could both be red)
so P(at least one red)= 2
5
17
12
16
5
17
4
16
35
5
, b) 2
68
17
12
17
5
17
5
17
145
289
11. a) n(F) = 3 ways (the only other possibility is both boys); n(E∩F) = 1; so P(E/F) = 1/3
b) n(F) = 2 (gb, gg); n(E∩F) = 1; so P(E/F) = ½
12. Binomial probability: C(5,2)
≈ 0.161
13. Use the complement, E’ = no kings; P(E’) =
14. P(E) =
≈ 0.6588; So P(E) = 0.3412
≈ 0.0011
15. a) n(F) = 30; n(E∩F) = 14 so P(E/F) = 14/30 = 7/15
b) n(F) = 186; n(E∩F) = 87 so P(E/F) = 87/186 = 29/62
c) n(F) = 205; n(E∩F) = 163 so P(E/F) = 163/205
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≈ .69.
16. These are dependent events. We want P(over 64 and Medicare) = P(over 64) ∙P(Medicare/ over 64)
= 0.363 ∙ 0.903 = 0.328
≈ 0.133
17. P(college grad/male) =
18. P(woman and woman becoming elem teacher) = 0.56 ∙ 0.317 = 0.178
19. P(defective and found defective) = 0.02 ∙ 0.92 = 0.0184 or 1.84%
≈ 0.187 or 18.7 %
20. P(no Ins given <$25000) =
21. Binomial experiment repeated 10 times. C(10,4)
22. 5 days: C(7,5)
7 days: C(7,7)
≈ 0.0384
6 days: C(7,6)
≈ 0.146
≈ 0.0064
≈ 0.00045 so P(> 4 days) ≈ 0.0453 or about 4.5 %
23. a) C(8,0) (.3)0 (.7)8 ≈ 0.0577
b) C(8,2) (.3)2 (.7)6 ≈ 0.2965
c) (Use complement) E’ : 0 undesirable or 1 undesirable or 2 undesirable
1 undesirable: C(8,1) (.3) (.7)7 ≈ 0.1977
P(E’) = 0.0577 + 0.2965 + 0.1977 =0.5519, so P(E) = 1- 0.5519 = 0.4481 or about 45%
24. Use complement: E’ no seeds germinate: C(4,0) (.75)0 (.25)4 = 0.0039
So P(E) = 0.996 or 99.6%
25. C(15,10)(.64)10 (.36)5 ≈ 0.209 or about 20.9%
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