9.6 Practice

9.1 Terminology

1. In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts a through j. (Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)

  • Let F be the event that a student is female.
  • Let M be the event that a student is male.
  • Let S be the event that a student has short hair.
  • Let L be the event that a student has long hair.
  1. The probability that a student does not have long hair.
  2. The probability that a student is male or has short hair.
  3. The probability that a student is a female and has long hair.
  4. The probability that a student is male, given that the student has long hair.
  5. The probability that a student has long hair, given that the student is male.
  6. Of all the female students, the probability that a student has short hair.
  7. Of all students with long hair, the probability that a student is female.
  8. The probability that a student is female or has long hair.
  9. The probability that a randomly selected student is a male student with short hair.
  10. The probability that a student is female.

Use the following information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, ten finger traps, and five bags of confetti. One party favor is chosen from the box at random.
Let H = the event of getting a hat.
Let N = the event of getting a noisemaker.
Let F = the event of getting a finger trap.
Let C = the event of getting a bag of confetti.

 

2. Find P(H).
3. Find P(N).
4. Find P(F).
5. Find P(C).


Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. One jelly bean is chosen from the box at random.
Let B = the event of getting a blue jelly bean
Let G = the event of getting a green jelly bean.
Let O = the event of getting an orange jelly bean.
Let P = the event of getting a purple jelly bean.
Let R = the event of getting a red jelly bean.
Let Y = the event of getting a yellow jelly bean.

 

6. Find P(B).
7. Find P(G).
8. Find P(P).
9. Find P(R).
10. Find P(Y).
11. Find P(O).


Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region).
Let A = the event that a country is in Asia.
Let E = the event that a country is in Europe.
Let F = the event that a country is in Africa.
Let N = the event that a country is in North America.
Let O = the event that a country is in Oceania.
Let S = the event that a country is in South America.

 

12. Find P(A).
13. Find P(E).
14. Find P(F).
15. Find P(N).
16. Find P(O).
17. Find P(S).
18. What is the probability of drawing a red card in a standard deck of 52 cards?
19. What is the probability of drawing a club in a standard deck of 52 cards?
20. What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six?
21. What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six?


Use the following information to answer the next two exercises. You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area.

Figure 3.10


Let B = the event of landing on blue.
Let R = the event of landing on red.
Let G = the event of landing on green.
Let Y = the event of landing on yellow.

 

22. If you land on Y, you get the biggest prize. Find P(Y).
23. If you land on red, you don’t get a prize. What is P(R)?


Use the following information to answer the next ten exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters.
Let I = the event that a player in an infielder.
Let O = the event that a player is an outfielder.
Let H = the event that a player is a great hitter.
Let N = the event that a player is not a great hitter.

 

24. Write the symbols for the probability that a player is not an outfielder.
25. Write the symbols for the probability that a player is an outfielder or is a great hitter.
26. Write the symbols for the probability that a player is an infielder and is not a great hitter.
27. Write the symbols for the probability that a player is a great hitter, given that the player is an infielder.
28. Write the symbols for the probability that a player is an infielder, given that the player is a great hitter.
29. Write the symbols for the probability that of all the outfielders, a player is not a great hitter.
30. Write the symbols for the probability that of all the great hitters, a player is an outfielder.
31. Write the symbols for the probability that a player is an infielder or is not a great hitter.
32. Write the symbols for the probability that a player is an outfielder and is a great hitter.
33. Write the symbols for the probability that a player is an infielder.
34. What is the word for the set of all possible outcomes?
35. What is conditional probability?

36. A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book

Let F = event that book is fiction
Let N = event that book is nonfiction
What is the sample space?

 

37. What is the sum of the probabilities of an event and its complement?


Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let E = the event that it lands on an even number. Let M = the event that it lands on a multiple of three.

38. What does P(E|M) mean in words?
39. What does P(E OR M) mean in words?

9.2 Independent and Mutually Exclusive Events

40. E and F are mutually exclusive events. P(E) = 0.4; P(F) = 0.5. Find P(E∣F).
41. J and K are independent events. P(J|K) = 0.3. Find P(J).

42. U and V are mutually exclusive events. P(U) = 0.26; P(V) = 0.37. Find:

  1. P(U AND V) =
  2. P(U|V) =
  3. P(U OR V) =
43. Q and R are independent events. P(Q) = 0.4 and P(Q AND R) = 0.1. Find P(R).

9.3 Two Basic Rules of Probability

Use the following information to answer the next ten exercises. Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. 37.6% of all Californians are Latino.

In this problem, let:

  • C = Californians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder.
  • L = Latino Californians

Suppose that one Californian is randomly selected.

44. Find P(C).
45. Find P(L).
46. Find P(C|L).
47. In words, what is C|L?
48. Find P(L AND C).
49. In words, what is L AND C?
50. Are L and C independent events? Show why or why not.
51. Find P(L OR C).
52. In words, what is L OR C?
53. Are L and C mutually exclusive events? Show why or why not.

9.4 Contingency Tables

Use the following information to answer the next four exercises. Table 3.15 shows a random sample of musicians and how they learned to play their instruments.

Gender Self-taught Studied in School Private Instruction Total
Female 12 38 22 72
Male 19 24 15 58
Total 31 62 37 130
Table 3.15
54. Find P(musician is a female).
55. Find P(musician is a male AND had private instruction).
56. Find P(musician is a female OR is self taught).
57. Are the events “being a female musician” and “learning music in school” mutually exclusive events?

9.5 Tree and Venn Diagrams

58. The probability that a man develops some form of cancer in his lifetime is 0.4567. The probability that a man 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. Let: C = a man develops cancer in his lifetime; P = man has at least one false positive. Construct a tree diagram of the situation.

9.6 Bring it together

Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African American people, 2,745 Native Hawaiian people, 12,831 Latino people, 8,378 Japanese people, and 7,650 White people. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African American people, 3,062 Native Hawaiian people, 4,932 Latino people, 10,680 Japanese people, and 9,877 White people. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African American people, 1,419 Native Hawaiian people, 1,406 Latino people, 4,715 Japanese people, and 6,062 White people. Of the people smoking at least 31 cigarettes per day, there were 759 African American people, 788 Native Hawaiian people, 800 Latino people, 2,305 Japanese people, and 3,970 White people.

59. Complete the table using the data provided.

Smoking Level African American Native Hawaiian Latino Japanese people White TOTALS
1–10
11–20
21–30
31+
TOTALS
Table 3.16 Smoking Levels by Ethnicity
60. Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.
61. Find the probability that the person was Latino.
62. In words, explain what it means to pick one person from the study who is “Japanese American AND smokes 21 to 30 cigarettes per day.” Also, find the probability.
63. In words, explain what it means to pick one person from the study who is “Japanese American OR smokes 21 to 30 cigarettes per day.” Also, find the probability.
64. In words, explain what it means to pick one person from the study who is “Japanese American GIVEN that person smokes 21 to 30 cigarettes per day.” Also, find the probability.
65. Prove that smoking level/day and ethnicity are dependent events.

Attribution

By Barbara Illowsky and Susan Dean, Introductory Statistics, Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction. Jun 23, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License CC BY-NC-SA 4.0. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

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