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Conditional Probability and Contingency Tables
Common Core: S.CP.4

Conditional Probability and Contingency Tables

Author: Sophia Tutorial
Description:

Calculate the conditional probability of an event from a contingency table.

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Tutorial
what's covered
This tutorial will cover conditional probability in the context of two-way tables. Our discussion breaks down as follows:

  1. Conditional Probability
  2. Tables


1. Conditional Probability

You can use two-way tables to find conditional probabilities. Recall that conditional probability is the probability that some event (B) occurs given that some other event (A) has already occurred. The probability of B given A is written this way:

P(B, given A) = P(B | A)

EXAMPLE

Suppose that 338 middle school students were asked which was their dominant hand. Here the results are shown in the two-way table:


Dominant Hand
Right Left Ambidextrous
Grade 6th 99 9 2 110
7th 90 31 0 121
8th 93 11 3 107


282 51 5 338

If a student is a sixth grader, what is the probability that he or she is left-handed? To find the answer, isolate the sixth grade row. This is a question of conditional probability: the probability of a student being left-handed given that the student is a sixth grader, P(L | 6). The formula looks like this:

P open parentheses L space vertical line space 6 close parentheses space equals space fraction numerator P open parentheses L space a n d space 6 close parentheses over denominator P open parentheses 6 close parentheses end fraction space equals space fraction numerator begin display style 9 over 338 end style over denominator begin display style 110 over 338 end style end fraction space equals space fraction numerator 9 space left parenthesis L e f t space h a n d e d space s i x t h minus g r a d e r s right parenthesis over denominator 110 space left parenthesis A l l space s i x t h minus g r a d e r s right parenthesis end fraction

This formula shows the probability of both left-handed and sixth grade divided by the probability of sixth grade. The probability that a student is left-handed and a sixth grader is 9 out of the 338 middle schoolers. The probability that a student is in sixth grade is 110 out of the 338. This reduces to 9 out of 110.

hint
Notice that you can use the probabilities 9/338 and 110/338, which were both divided by 338, the grand total. Or, you can just use the frequency from the cell for both left-handed and sixth, 9, and from the marginal distribution in the row totals for sixth grade, 110.

P left parenthesis 6 space vertical line space L right parenthesis equals fraction numerator P left parenthesis 6 space a n d space L right parenthesis over denominator P left parenthesis L right parenthesis end fraction
The numerator would be the same as the first question; we need to find the probability of both left-handed and sixth grade. However, the denominator is different. In this question, we need to find the probability of a student being left-handed.
P left parenthesis 6 space vertical line thin space L right parenthesis equals fraction numerator P left parenthesis 6 space a n d space L right parenthesis over denominator P left parenthesis L right parenthesis end fraction equals 9 over 51

With contingency tables, you can just look at the corresponding cells and column or row totals. For the probability of left-handed and sixth grade, the frequency was 9. For the probability of left-handed, there were 51 of the lefties. So, the answer is 9 out of 51.

try it
Using the table above, find the following probabilities.

Questions Written as Conditional Probability Conditional Probability Formula
What is the probability that a seventh-grade student is ambidextrous? What is the probability that a student is ambidextrous, given they are a seventh grader? P left parenthesis A space vertical line thin space 7 right parenthesis equals fraction numerator P left parenthesis A space a n d space 7 right parenthesis over denominator P left parenthesis 7 right parenthesis end fraction equals 0 over 121 equals 0
What is the probability that a student who is right-handed is in eighth grade? What is the probability that a student is in eight grade, given that he or she is right-handed? P left parenthesis 8 space vertical line thin space R right parenthesis equals fraction numerator P left parenthesis 8 space a n d space R right parenthesis over denominator P left parenthesis R right parenthesis end fraction equals 93 over 282


2. Tables

You can use a two-way table that actually has probabilities in it or relative frequencies.

EXAMPLE

A class of 10th graders was asked if they prefer cheese, pepperoni, or sausage pizza. The percentages are shown below: 41% of all of these kids are girls that enjoy cheese pizza, 12% of all of the kids are boys that enjoy pepperoni, etc.


Cheese Pepperoni Sausage
Boy 0.05 0.12 0.19 0.36
Girl 0.41 0.16 0.07 0.64

0.46 0.28 0.26 1

Given that the student is a boy, what is the probability that he likes cheese?
To find the probability of a student preferring cheese pizza given that he's a boy, you can use the probabilities from the marginal distributions in the column or row totals, instead of the frequencies.
P left parenthesis C space vertical line thin space B right parenthesis equals fraction numerator P left parenthesis C space a n d space B right parenthesis over denominator P left parenthesis B right parenthesis end fraction equals fraction numerator 0.05 over denominator 0.36 end fraction equals 0.139
Therefore, the probability that a student enjoys cheese and is a boy is the 0.05 value from the table. The probability of the student being a boy, in this particular sample, is 36%. This reveals that there's about 14% probability that if you are a boy, you'll prefer cheese pizza.


summary
Conditional probability is the probability that some event (Event B) follows some other event which has already occurred (Event A). It's calculated by dividing the joint probability-- the probability of A and B--by the probability of the event which has already occurred. This formula works for all events, not just for independent events or mutually exclusive events, and the data for these formulas can be found in two-way tables.

Good luck!

Source: Adapted from Sophia tutorial by Jonathan Osters.