Probability Topics: Contingency Tables

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1 Connexions module: m835 Probability Topics: Contingency Tables Susan Dean Barbara Illowsky, Ph.D. This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract This module introduces the contingency table as a way of determining conditional probabilities. A contingency table provides a dierent way of calculating probabilities. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two dierent variables that may be dependent or contingent on one another. Later on, we will use contingency tables again, but in another manner. Example Suppose a study of speeding violations and drivers who use car phones produced the following ctional data: Speeding violation in the last year No speeding violation in the last year Car phone user Not a car phone user Total Table Total The total number of people in the sample is. The row totals are 305 and 50. The column totals are 70 and 85. Notice that = and =. Calculate the following probabilities using the table Problem P(person is a car phone user) = number of car phone users total number in study = 305 Version.: Feb 2, 20 :09 am US/Central Source URL: Saylor URL: Page of 5

2 Connexions module: m835 2 Problem 2 P(person had no violation in the last year) = number that had no violation total number in study = 85 Problem 3 P(person had no violation in the last year AND was a car phone user) = 280 Problem P(person is a car phone user OR person had no violation in the last year) = ( ) = 70 Problem 5 P(person is a car phone user GIVEN person had a violation in the last year) = (The sample space is reduced to the number of persons who had a violation.) Problem P(person had no violation last year GIVEN person was not a car phone user) = (The sample space is reduced to the number of persons who were not car phone users.) Example 2 The following table shows a random sample of 00 hikers and the areas of hiking preferred: Hiking Area Preference Sex The Coastline Near Lakes and Streams On Mountain Peaks Total Female 8 5 Male 55 Total Table 2 Problem ( on p. 5.) Complete the table. Problem 2 ( on p. 5.) Are the events "being female" and "preferring the coastline" independent events? Let F = being female and let C = preferring the coastline. Source URL: Saylor URL: Page 2 of 5

3 Connexions module: m835 3 a. P(F AND C) = b. 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. Problem 3 ( on p. 5.) 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. a. What word tells you this is a conditional? b. Fill in the blanks and calculate the probability: P( ) =. c. Is the sample space for this problem all 00 hikers? If not, what is it? Problem ( on p. 5.) Find the probability that a person is female or prefers hiking on mountain peaks. Let F = being female and let P = prefers mountain peaks. a. P(F) = b. P(P) = c. P(F AND P) = d. Therefore, P(F OR P) = Example 3 Muddy Mouse lives in a cage with 3 doors. If Muddy goes out the rst door, the probability that he gets caught by Alissa the cat is 5 and the probability he is not caught is 5. If he goes out the second door, the probability he gets caught by Alissa is and the probability he is not caught is 3. The probability that Alissa catches Muddy coming out of the third door is 2 and the probability she does not catch Muddy is 2. It is equally likely that Muddy will choose any of the three doors so the probability of choosing each door is 3. Door Choice Caught or Not Door One Door Two Door Three Total Caught Not Caught 5 2 Total Table 3 The rst entry 5 = ( ( 5) 3) is P(Door One AND Caught). The entry 5 = ( ( 5) 3) is P(Door One AND Not Caught). Verify the remaining entries. Problem ( on p. 5.) Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is. Problem 2 What is the probability that Alissa does not catch Muddy? Source URL: Saylor URL: Page 3 of 5

4 Connexions module: m835 0 Problem 3 What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa? 9 9 note: You could also do this problem by using a probability tree. See the Tree Diagrams (Optional) section of this chapter for examples. "Probability Topics: Tree Diagrams (optional)" < Source URL: Saylor URL: Page of 5

5 Connexions module: m835 5 s to Exercises in this Module to Example 2, Problem (p. 2) Hiking Area Preference Sex The Coastline Near Lakes and Streams On Mountain Peaks Total Female 8 5 Male Total to Example 2, Problem 2 (p. 2) Table a. P(F AND C) = 8 00 = 0.8 b. P (F) P (C) = = = 0.53 P(F AND C) P (F) P (C), so the events F and C are not independent. to Example 2, Problem 3 (p. 3) a. The word 'given' tells you that this is a conditional. b. P(M L) = 25 c. No, the sample space for this problem is. to Example 2, Problem (p. 3) a. P(F) = 5 00 b. P(P) = c. P(F AND P) = 00 d. P(F OR P) = = to Example 3, Problem (p. 3) Door Choice Caught or Not Door One Door Two Door Three Total Caught 5 Not Caught 5 5 Total Table Glossary Denition : Contingency Table The method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other. The table provides an easy way to calculate conditional probabilities. Source URL: Saylor URL: Page 5 of 5

Lesson 3: Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables

Lesson 3: Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables Classwork Example 1 Students at Rufus King High School were discussing some of the challenges of finding space for