Probabilities and Contingency Tables

Learning Outcomes

  • Calculate probabilities for events that are mutually exclusive and not mutually exclusive for a given contingency table
  • Calculate probabilities for independent events for a given contingency table
  • Calculate conditional probabilities for a given contingency table
  • Determine if two events are independent for a given contingency table

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.

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 [latex]25[/latex] [latex]280[/latex] [latex]305[/latex]
Not a cell phone user [latex]45[/latex] [latex]405[/latex] [latex]450[/latex]
Total [latex]70[/latex] [latex]685[/latex] [latex]755[/latex]

The total number of people in the sample is [latex]755[/latex]. The row totals are [latex]305[/latex] and [latex]450[/latex]. The column totals are [latex]70[/latex] and [latex]685[/latex]. Notice that [latex]305 + 450 = 755 \text{ and } 70 + 685 = 755[/latex].

Calculate the following probabilities using the table.

  1. Find [latex]P[/latex](Person is a car phone user).
  2. Find [latex]P[/latex](Person had no violation in the last year).
  3. Find [latex]P[/latex](Person had no violation in the last year AND was a car phone user).
  4. Find [latex]P[/latex](Person is a car phone user OR person had no violation in the last year).
  5. Find [latex]P[/latex](Person is a car phone user GIVEN person had a violation in the last year).
  6. Find [latex]P[/latex](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 [latex]55[/latex] [latex]295[/latex] [latex]350[/latex]
Does not stretch [latex]231[/latex] [latex]219[/latex] [latex]450[/latex]
Total [latex]286[/latex] [latex]514[/latex] [latex]800[/latex]
  1. What is [latex]P[/latex](athlete stretches before exercising)?
  2. What is [latex]P[/latex](athlete stretches before exercising|no injury in the last year)?

Example

This table shows a random sample of [latex]100[/latex] hikers and the areas of hiking they prefer.

Hiking Area Preference

Sex The Coastline Near Lakes and Streams On Mountain Peaks Total
Female [latex]18[/latex] [latex]16[/latex] ___ [latex]45[/latex]
Male ___ ___ [latex]14[/latex] [latex]55[/latex]
Total ___ [latex]41[/latex] ___ ___
  1. Complete the table.
  2. Are the events “being female” and “preferring the coastline” independent events? Let [latex]F[/latex] = being female and let [latex]C[/latex] = preferring the coastline.
    1. Find [latex]P(F \text{ AND } C)[/latex].
    2. Find [latex]P(F)P(C)[/latex].

    Are these two numbers the same? If they are, then [latex]F[/latex] and [latex]C[/latex] are independent. If they are not, then [latex]F[/latex] and [latex]C[/latex] are not independent.

  3. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let [latex]M[/latex] = being male, and let [latex]L[/latex] = prefers hiking near lakes and streams.
    1. What word tells you this is a conditional?
    2. Fill in the blanks and calculate the probability: [latex]P[/latex](___|___) = ___.
    3. Is the sample space for this problem all [latex]100[/latex] 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 [latex]P(F)[/latex].
    2. Find [latex]P(P)[/latex].
    3. Find [latex]P(F \text{ AND } P)[/latex].
    4. Find [latex]P(F \text{ OR } P)[/latex].

try it

This table shows a random sample of [latex]200[/latex] cyclists and the routes they prefer. Let [latex]M[/latex] = males and [latex]H[/latex] = hilly path.

Gender Lake Path Hilly Path Wooded Path Total
Female [latex]45[/latex] [latex]38[/latex] [latex]27[/latex] [latex]110[/latex]
Male [latex]26[/latex] [latex]52[/latex] [latex]12[/latex] [latex]90[/latex]
Total [latex]71[/latex] [latex]90[/latex] [latex]39[/latex] [latex]200[/latex]
  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 [latex]\displaystyle\frac{{1}}{{5}}[/latex] and the probability he is not caught is [latex]\displaystyle\frac{{4}}{{5}}[/latex]. If he goes out the second door, the probability he gets caught by Alissa is [latex]\displaystyle\frac{{1}}{{4}}[/latex] and the probability he is not caught is [latex]\displaystyle\frac{{3}}{{4}}[/latex]. The probability that Alissa catches Muddy coming out of the third door is [latex]\displaystyle\frac{{1}}{{2}}[/latex] and the probability she does not catch Muddy is [latex]\displaystyle\frac{{1}}{{2}}[/latex]. It is equally likely that Muddy will choose any of the three doors so the probability of choosing each door is [latex]\displaystyle\frac{{1}}{{3}}[/latex].

Door Choice

Caught or Not Door One Door Two Door Three Total
Caught [latex]\displaystyle\frac{{1}}{{15}}[/latex] [latex]\displaystyle\frac{{1}}{{12}}[/latex] [latex]\displaystyle\frac{{1}}{{6}}[/latex] ____
Not Caught [latex]\displaystyle\frac{{4}}{{15}}[/latex] [latex]\displaystyle\frac{{3}}{{12}}[/latex] [latex]\displaystyle\frac{{1}}{{6}}[/latex] ____
Total ____ ____ ____ [latex]1[/latex]
  • The first entry [latex]\displaystyle\frac{{1}}{{15}}={(\frac{{1}}{{5}})}{(\frac{{1}}{{3}})}[/latex] is [latex]P[/latex](Door One AND Caught)
  • The entry [latex]\displaystyle\frac{{4}}{{15}}={(\frac{{4}}{{5}})}{(\frac{{1}}{{3}})}[/latex] is [latex]P[/latex](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 [latex]1[/latex].
  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 [latex]100,000[/latex] inhabitants from 2008 to 2011 in the U.S.

Year Robbery Burglary Rape Vehicle Total
2008 [latex]145.7[/latex] [latex]732.1[/latex] [latex]29.7[/latex] [latex]314.7[/latex]
2009 [latex]133.1[/latex] [latex]717.7[/latex] [latex]29.1[/latex] [latex]259.2[/latex]
2010 [latex]119.3[/latex] [latex]701[/latex] [latex]27.7[/latex] [latex]239.1[/latex]
2011 [latex]113.7[/latex] [latex]702.2[/latex] [latex]26.8[/latex] [latex]229.6[/latex]
Total

 

TOTAL each column and each row. Total data = [latex]4,520.7[/latex]

  1. Find [latex]P[/latex](2009 AND Robbery).
  2. Find [latex]P[/latex](2010 AND Burglary).
  3. Find [latex]P[/latex](2010 OR Burglary).
  4. Find [latex]P[/latex](2011|Rape).
  5. Find [latex]P[/latex](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 Total
Obese [latex]18[/latex] [latex]28[/latex] [latex]14[/latex]
Normal [latex]20[/latex] [latex]51[/latex] [latex]28[/latex]
Underweight [latex]12[/latex] [latex]25[/latex] [latex]9[/latex]
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?