Contingency Tables

Learning Outcomes

  • Construct and interpret Contingency Tables


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.

The following video shows and example of finding the probability of an event from a table.



Example

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
2525
280280
305305
Not a cell phone user
4545
405405
450450
Total
7070
685685
755755
The total number of people in the sample is
755755
. The row totals are
305305
and
450450
. The column totals are
7070
and
685685
. Notice that
305+450=755 and 70+685=755305 + 450 = 755 \text{ and } 70 + 685 = 755
.

Calculate the following probabilities using the table.

  1. Find
    PP
    (Person is a car phone user).
  2. Find
    PP
    (person had no violation in the last year).
  3. Find
    PP
    (Person had no violation in the last year AND was a car phone user).
  4. Find
    PP
    (Person is a car phone user OR person had no violation in the last year).
  5. Find
    PP
    (Person is a car phone user GIVEN person had a violation in the last year).
  6. Find
    PP
    (Person had no violation last year GIVEN person was not a car phone user)






 


This video shows an example of how to determine the probability of an AND event using a contingency table.



try it

This table 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
5555
295295
350350
Does not stretch
231231
219219
450450
Total
286286
514514
800800
  1. What is
    PP
    (athlete stretches before exercising)?
  2. What is
    PP
    (athlete stretches before exercising|no injury in the last year)?






Example

This table shows a random sample of
100100
hikers and the areas of hiking they prefer.

Hiking Area Preference

Sex The Coastline Near Lakes and Streams On Mountain Peaks Total
Female
1818
1616
___
4545
Male ___ ___
1414
5555
Total ___
4141
___ ___
  1. Complete the table.
  2. Are the events "being female" and "preferring the coastline" independent events?Let 
    FF
    = being female and let
    CC
    = preferring the coastline.

    1. Find
      P(F AND C)P(F \text{ AND } C)
      .
    2. Find
      P(F)P(C)P(F)P(C)


    Are these two numbers the same? If they are, then
    FF
    and
    CC
    are independent. If they are not, then
    FF
    and
    CC
    are not independent.
  3. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let
    MM
    = being male, and let
    LL
    = prefers hiking near lakes and streams.

    1. What word tells you this is a conditional?
    2. Fill in the blanks and calculate the probability:
      PP
      (___|___) = ___.
    3. Is the sample space for this problem all
      100100
      hikers? If not, what is it?


  4. 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)P(F)
      .
    2. Find
      P(P)P(P)
      .
    3. Find
      P(F AND P)P(F \text{ AND } P)
      .
    4. Find
      P(F OR P)P(F \text{ OR } P)
      .








try it

This table shows a random sample of
200200
cyclists and the routes they prefer. Let 
MM
= males and
HH
= hilly path.

Gender Lake Path Hilly Path Wooded Path Total
Female
4545
3838
2727
110110
Male
2626
5252
1212
9090
Total
7171
9090
3939
200200
  1. Out of the males, what is the probability that the cyclist prefers a hilly path?
  2. Are the events "being male" and "preferring the hilly path" independent events?






Example

Muddy Mouse lives in a cage with three doors. If Muddy goes out the first door, the probability that he gets caught by Alissa the cat is 
15\displaystyle\frac{{1}}{{5}}
.

Door Choice

Caught or Not Door One Door Two Door Three Total
Caught
115\displaystyle\frac{{1}}{{15}}
112\displaystyle\frac{{1}}{{12}}
16\displaystyle\frac{{1}}{{6}}
____
Not Caught
415\displaystyle\frac{{4}}{{15}}
312\displaystyle\frac{{3}}{{12}}
16\displaystyle\frac{{1}}{{6}}
____
Total ____ ____ ____
11
  • The first entry
    115=(15)(13)\displaystyle\frac{{1}}{{15}}={(\frac{{1}}{{5}})}{(\frac{{1}}{{3}})}
    is
    PP
    (Door One AND Caught)
  • The entry
    415=(45)(13)\displaystyle\frac{{4}}{{15}}={(\frac{{4}}{{5}})}{(\frac{{1}}{{3}})}
    is
    PP
    (Door One AND Not Caught)


Verify the remaining entries.

  1. Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is
    11
    .
  2. What is the probability that Alissa does not catch Muddy?
  3. What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa?






 

example

This table contains the number of crimes per
100,000100,000
inhabitants from 2008 to 2011 in the U.S.

United States Crime Index Rates Per
100,000100,000
Inhabitants 2008–2011

Year Robbery Burglary Rape Vehicle Total
2008
145.7145.7
732.1732.1
29.729.7
314.7314.7
2009
133.1133.1
717.7717.7
29.129.1
259.2259.2
2010
119.3119.3
701701
27.727.7
239.1239.1
2011
113.7113.7
702.2702.2
26.826.8
229.6229.6
Total
TOTAL each column and each row. Total data =
4,520.74,520.7


  1. Find
    PP
    (2009 AND Robbery).
  2. Find
    PP
    (2010 AND Burglary).
  3. Find
    PP
    (2010 OR Burglary).
  4. Find
    PP
    (2011|Rape).
  5. Find
    PP
    (Vehicle|2008).






 


This video gives and example of determining an "OR" probability given a table.



try it

This table relates the weights and heights of a group of individuals participating in an observational study.

Weight/Height Tall Medium Short Totals
Obese
1818
2828
1414
Normal
2020
5151
2828
Underweight
1212
2525
99
Totals
  1. Find the total for each row and column
  2. Find the probability that a randomly chosen individual from this group is Tall.
  3. Find the probability that a randomly chosen individual from this group is Obese and Tall.
  4. Find the probability that a randomly chosen individual from this group is Tall given that the idividual is Obese.
  5. Find the probability that a randomly chosen individual from this group is Obese given that the individual is Tall.
  6. Find the probability a randomly chosen individual from this group is Tall and Underweight.
  7. Are the events Obese and Tall independent?






 

References

"Blood Types." American Red Cross, 2013. Available online at http://www.redcrossblood.org/learn-about-blood/blood-types (accessed May 3, 2013).

Data from the National Center for Health Statistics, part of the United States Department of Health and Human Services.

Data from United States Senate. Available online at www.senate.gov (accessed May 2, 2013).

Haiman, Christopher A., Daniel O. Stram, Lynn R. Wilkens, Malcom C. Pike, Laurence N. Kolonel, Brien E. Henderson, and Loīc Le Marchand. "Ethnic and Racial Differences in the Smoking-Related Risk of Lung Cancer." The New England Journal of Medicine, 2013. Available online at http://www.nejm.org/doi/full/10.1056/NEJMoa033250 (accessed May 2, 2013).

"Human Blood Types." Unite Blood Services, 2011. Available online at http://www.unitedbloodservices.org/learnMore.aspx (accessed May 2, 2013).

Samuel, T. M. "Strange Facts about RH Negative Blood." eHow Health, 2013. Available online at http://www.ehow.com/facts_5552003_strange-rh-negative-blood.html (accessed May 2, 2013).

"United States: Uniform Crime Report – State Statistics from 1960–2011." The Disaster Center. Available online at http://www.disastercenter.com/crime/ (accessed May 2, 2013).

Concept 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 probabilites that have multiple dependent variables.

Licenses and Attributions

More Study Resources for You

Show More