Example \(\PageIndex{2}\)
Table shows a random sample of 100 hikers and the areas of hiking they prefer.
Hiking Area Preference
| Sex |
The Coastline |
Near Lakes and Streams |
On Mountain Peaks |
Total |
| Female |
18 |
16 |
___ |
45 |
| Male |
___ |
___ |
14 |
55 |
| Total |
___ |
41 |
___ |
___ |
- Complete the table.
- Are the events "being female" and "preferring the coastline" independent events? Let F = being female and let C = preferring the coastline.
- Find P(F AND C).
- Find P(F)P(C)
- Are these two numbers the same? If they are, then F and C are independent. If they are not, then F and C are not independent.
- Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let \(\text{M} =\) being male, and let \(\text{L} =\) prefers hiking near lakes and streams.
- What word tells you this is a conditional?
- Fill in the blanks and calculate the probability: \(P\)(___|___) = ___.
- Is the sample space for this problem all 100 hikers? If not, what is it?
- Find the probability that a person is female or prefers hiking on mountain peaks. Let \(\text{F} =\) being female, and let \(\text{P} =\) prefers mountain peaks.
- Find \(P(\text{F})\).
- Find \(P(\text{P})\).
- Find \(P(\text{F AND P})\).
- Find \(P(\text{F OR P})\).
Answers
a.
Hiking Area Preference
| Sex |
The Coastline |
Near Lakes and Streams |
On Mountain Peaks |
Total |
| Female |
18 |
16 |
11 |
45 |
| Male |
16 |
25 |
14 |
55 |
| Total |
34 |
41
|
25 |
100 |
b.
\(P(\text{F AND C}) = \dfrac{18}{100} = 0.18\)
\(P(\text{F})P(\text{C}) = \left(\dfrac{45}{100}\right) \left(\dfrac{34}{100}\right) = (0.45)(0.34) = 0.153\)
\(P(\text{F AND C}) \neq P(\text{F})P(\text{C})\), so the events \(\text{F}\) and \(\text{C}\) are not independent.
c.
- The word 'given' tells you that this is a conditional.
- \(P(\text{M|L}) = \dfrac{25}{41}\)
- No, the sample space for this problem is the 41 hikers who prefer lakes and streams.
d.
- Find \(P(\text{F})\).
- Find \(P(\text{P})\).
- Find \(P(\text{F AND P})\).
- Find \(P(\text{F OR P})\).
d.
- \(P(\text{F}) = \dfrac{45}{100}\)
- \(P(\text{P}) = \dfrac{25}{100}\)
- \(P(\text{F AND P}) = \dfrac{11}{100}\)
- \(P(\text{F OR P}) = \dfrac{45}{100} + \dfrac{25}{100} - \dfrac{11}{100} = \dfrac{59}{100}\)
Example \(\PageIndex{4}\)
Table contains the number of crimes per 100,000 inhabitants from 2008 to 2011 in the U.S.
United States Crime Index Rates Per 100,000 Inhabitants 2008–2011
| Year |
Robbery |
Burglary |
Rape |
Vehicle |
Total |
| 2008 |
145.7 |
732.1 |
29.7 |
314.7 |
|
| 2009 |
133.1 |
717.7 |
29.1 |
259.2 |
|
| 2010 |
119.3 |
701 |
27.7 |
239.1 |
|
| 2011 |
113.7 |
702.2 |
26.8 |
229.6 |
|
| Total |
|
|
|
|
|
TOTAL each column and each row. Total data = 4,520.7
- Find \(P(\text{2009 AND Robbery})\).
- Find \(P(\text{2010 AND Burglary})\).
- Find \(P(\text{2010 OR Burglary})\).
- Find \(P(\text{2011|Rape})\).
- Find \(P(\text{Vehicle|2008})\).
Answer
a. 0.0294, b. 0.1551, c. 0.7165, d. 0.2365, e. 0.2575
Review
There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables help display data and are particularly useful when calculating probabilities that have multiple dependent variables.
Use the following information to answer the next four exercises. Table 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 |
Exercise 3.5.4
Find P(musician is a female).
Exercise 3.5.5
Find \(P(\text{musician is a male AND had private instruction})\).
Answer
\(P(\text{musician is a male AND had private instruction}) = \dfrac{15}{130} = \dfrac{3}{26} = 0.12\)
Exercise 3.5.6
Find P(musician is a female OR is self taught).
Exercise 3.5.7
Are the events “being a female musician” and “learning music in school” mutually exclusive events?
Answer
The events are not mutually exclusive. It is possible to be a female musician who learned music in school.
Bringing 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 Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites.
Exercise 3.5.8
Complete the table using the data provided. Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.
Smoking Levels by Ethnicity
| Smoking Level |
African American |
Native Hawaiian |
Latino |
Japanese Americans |
White |
TOTALS |
| 1–10 |
|
|
|
|
|
|
| 11–20 |
|
|
|
|
|
|
| 21–30 |
|
|
|
|
|
|
| 31+ |
|
|
|
|
|
|
| TOTALS |
|
|
|
|
|
|
Exercise 3.5.9
Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.
Answer
\(\dfrac{35,065}{100,450}\)
Exercise 3.5.10
Find the probability that the person was Latino.
Exercise 3.5.11
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.
Answer
To pick one person from the study who is Japanese American AND smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes. The sample space should include everyone in the study. The probability is \(\dfrac{4,715}{100,450}\).
Exercise 3.5.12
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.
Exercise 3.5.13
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.
Answer
To pick one person from the study who is Japanese American given that person smokes 21-30 cigarettes per day, means that the person must fulfill both criteria and the sample space is reduced to those who smoke 21-30 cigarettes per day. The probability is \(\dfrac{4,715}{15,273}\).
Exercise 3.5.14
Prove that smoking level/day and ethnicity are dependent events.
