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3.4 Contingency tables and probability trees

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Contingency tables

A contingency table provides a way of portraying data that can facilitate calculating probabilities. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. Later on, we will use contingency tables again, but in another manner.

Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data:

Speeding violation in the last year No speeding violation in the last year Total
Cell phone user 25 280 305
Not a cell phone user 45 405 450
Total 70 685 755

The total number of people in the sample is 755. The row totals are 305 and 450. The column totals are 70 and 685. Notice that 305 + 450 = 755 and 70 + 685 = 755.

Calculate the following probabilities using the table.

a. Find P (Person is a car phone user).

a. number of car phone users total number in study   =   305 755

b. Find P (person had no violation in the last year).

b. number that had no violation total number in study   =   685 755

c. Find P (Person had no violation in the last year ∩ was a car phone user).

c. 280 755

d. Find P (Person is a car phone user ∪ person had no violation in the last year).

d. ( 305 755   +   685 755 )   −   280 755   =   710 755

e. Find P (Person is a car phone user | person had a violation in the last year).

e. 25 70 (The sample space is reduced to the number of persons who had a violation.)

f. Find P (Person had no violation last year | person was not a car phone user)

f. 405 450 (The sample space is reduced to the number of persons who were not car phone users.)

Try it

[link] shows the number of athletes who stretch before exercising and how many had injuries within the past year.

Injury in last year No injury in last year Total
Stretches 55 295 350
Does not stretch 231 219 450
Total 286 514 800

  1. What is P (athlete stretches before exercising)?
  2. What is P (athlete stretches before exercising | no injury in the last year)?
  1. P (athlete stretches before exercising) = 350 800 = 0.4375
  2. P (athlete stretches before exercising | no injury in the last year) = 295 514 = 0.5739

[link] 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 ___ ___

a. Complete the table.

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. Are the events "being female" and "preferring the coastline" independent events?

Let F = being female and let C = preferring the coastline.

  1. Find P ( F ∩ C ) .
  2. 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.

b.

  1. P ( F ∩ C ) = 18 100 = 0.18
  2. P ( F ) P ( C ) = ( 45 100 ) ( 34 100 ) = (0.45)(0.34) = 0.153

P ( F ∩ C ) ≠ P ( F ) P ( C ), so the events F and C are not independent.

c. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let M = being male, and let L = prefers hiking near lakes and streams.

  1. What word tells you this is a conditional?
  2. Fill in the blanks and calculate the probability: P (___ | ___) = ___.
  3. Is the sample space for this problem all 100 hikers? If not, what is it?

c.

  1. The word 'given' tells you that this is a conditional.
  2. P ( M | L ) = 25 41
  3. No, the sample space for this problem is the 41 hikers who prefer lakes and streams.

d. Find the probability that a person is female or prefers hiking on mountain peaks. Let F = being female, and let P = prefers mountain peaks.

  1. Find P ( F ).
  2. Find P ( P ).
  3. Find P ( F ∩ P ) .
  4. Find P ( F ∪ P ) .

d.

  1. P ( F ) = 45 100
  2. P ( P ) = 25 100
  3. P ( F ∩ P ) = 11 100
  4. P ( F ∪ P ) = 45 100 + 25 100 - 11 100 = 59 100

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Source:  OpenStax, Introductory statistics. OpenStax CNX. Aug 09, 2016 Download for free at http://legacy.cnx.org/content/col11776/1.26
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