Module 3: Probability
Contingency Tables
Barbara Illowsky & OpenStax et al.
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 | 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.
- Find P(Person is a car phone user).
- Find P(person had no violation in the last year).
- Find P(Person had no violation in the last year AND was a car phone user).
- Find P(Person is a car phone user OR person had no violation in the last year).
- Find P(Person is a car phone user GIVEN person had a violation in the last year).
- Find P(Person had no violation last year GIVEN person was not a car phone user)
Solution:
- [latex]displaystylefrac{{text{number of car phone users}}}{{text{total number in study}}}=frac{{305}}{{755}}[/latex]
- [latex]displaystylefrac{{text{number that had no violation}}}{{text{total number in study}}}=frac{{685}}{{755}}[/latex]
- [latex]displaystylefrac{{280}}{{755}}[/latex]
- [latex]displaystyle{(frac{{305}}{{755}}+frac{{685}}{{755}})}-frac{{280}}{{755}}=frac{{710}}{{755}}[/latex]
- [latex]displaystylefrac{{25}}{{70}}[/latex](The sample space is reduced to the number of persons who had a violation.)
- [latex]displaystylefrac{{405}}{{450}}[/latex] (The sample space is reduced to the number of persons who were not car phone users.)
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 | 55 | 295 | 350 |
| Does not stretch | 231 | 219 | 450 |
| Total | 286 | 514 | 800 |
- What is P(athlete stretches before exercising)?
- What is P(athlete stretches before exercising|no injury in the last year)?
- P(athlete stretches before exercising) = [latex]displaystylefrac{{350}}{{800}}[/latex] = 0.4375
- P(athlete stretches before exercising|no injury in the last year) = [latex]displaystylefrac{{295}}{{514}}[/latex] = 0.5739
Example
This table shows a random sample of 100 hikers and the areas of hiking t