3.4 Contingency tables

Collaborative statistics Page 1 / 2

This module introduces the contingency table as a way of determining conditional probabilities.

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. Contingincy tables provide a way of portraying data that can facilitate calculating probabilities.

Suppose a study of speeding violations and drivers who use car phones produced the following fictional data:

	Speeding violation in the last year	No speeding violation in the last year	Total
Car phone user	25	280	305
Not a car 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

$P(person is a car phone user) =$

$\frac{number of car phone users}{total number in study} = \frac{305}{755}$

Got questions? Get instant answers now!

$P(person had no violation in the last year) =$

$\frac{number that had no violation}{total number in study} = \frac{685}{755}$

Got questions? Get instant answers now!

$P(person had no violation in the last year AND was a car phone user) =$

$\frac{280}{755}$

Got questions? Get instant answers now!

$P(person is a car phone user OR person had no violation in the last year) =$

$(\frac{305}{755} + \frac{685}{755}) - \frac{280}{755} = \frac{710}{755}$

Got questions? Get instant answers now!

$P(person is a car phone user GIVEN person had a violation in the last year) =$

$\frac{25}{70}$ (The sample space is reduced to the number of persons who had a violation.)

Got questions? Get instant answers now!

$P(person had no violation last year GIVEN person was not a car phone user) =$

$\frac{405}{450}$ (The sample space is reduced to the number of persons who were not car phone users.)

Got questions? Get instant answers now!

The following table shows a random sample of 100 hikers and the areas of hiking preferred:

Hiking area preference
Sex	The Coastline	Near Lakes and Streams	On Mountain Peaks	Total
Female	18	16	___	45
Male	___	___	14	55
Total	___	41	___	___

Complete the table.

Hiking area preference
Sex	The Coastline	Near Lakes and Streams	On Mountain Peaks	Total
Female	18	16	11	45
Male	16	25	14	55
Total	34	41	25	100

Got questions? Get instant answers now!

Are the events "being female" and "preferring the coastline" independent events?

Let $F$ = being female and let $C$ = preferring the coastline.

a $P(F AND C)$ =
b $P(F) \cdot P(C)$ =

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.

a $P(F AND C) = \frac{18}{100} = 0.18$
b $P(F) \cdot P(C) = \frac{45}{100} \cdot \frac{34}{100} = 0.45 \cdot 0.34 = 0.153$

$P(F AND C) \neq P(F) \cdot P(C)$ , so the events $F$ and $C$ are not independent.

Got questions? Get instant answers now!

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 100 hikers? If not, what is it?

a The word 'given' tells you that this is a conditional.
b $P(M|L) = \frac{25}{41}$
c No, the sample space for this problem is 41.

Got questions? Get instant answers now!

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, $P(F OR P) =$

a $P(F) = \frac{45}{100}$
b $P(P) = \frac{25}{100}$
c $P(F AND P) = \frac{11}{100}$
d $P(F OR P) = \frac{45}{100} + \frac{25}{100} - \frac{11}{100} = \frac{59}{100}$

Got questions? Get instant answers now!

Muddy Mouse lives in a cage with 3 doors. If Muddy goes out the first door, the probability that he gets caught by Alissa the cat is $\frac{1}{5}$ and the probability he is not caught is $\frac{4}{5}$ . If he goes out the second door, the probability he gets caught by Alissa is $\frac{1}{4}$ and the probability he is not caught is $\frac{3}{4}$ . The probability that Alissa catches Muddy coming out of the third door is $\frac{1}{2}$ and the probability she does not catch Muddy is $\frac{1}{2}$ . It is equally likely that Muddy will choose any of the three doors so the probability of choosing each door is $\frac{1}{3}$ .

Door choice
Caught or Not	Door One	Door Two	Door Three	Total
Caught	$\frac{1}{15}$	$\frac{1}{12}$	$\frac{1}{6}$	____
Not Caught	$\frac{4}{15}$	$\frac{3}{12}$	$\frac{1}{6}$	____
Total	____	____	____	1

The first entry $\frac{1}{15} = (\frac{1}{5}) (\frac{1}{3})$ is $P(Door One AND Caught)$ .
The entry $\frac{4}{15} = (\frac{4}{5}) (\frac{1}{3})$ is $P(Door One AND Not Caught)$ .

Verify the remaining entries.

Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1.

Door choice
Caught or Not	Door One	Door Two	Door Three	Total
Caught	$\frac{1}{15}$	$\frac{1}{12}$	$\frac{1}{6}$	$\frac{19}{60}$
Not Caught	$\frac{4}{15}$	$\frac{3}{12}$	$\frac{1}{6}$	$\frac{41}{60}$
Total	$\frac{5}{15}$	$\frac{4}{12}$	$\frac{2}{6}$	1

Got questions? Get instant answers now!

What is the probability that Alissa does not catch Muddy?

$\frac{41}{60}$

Got questions? Get instant answers now!

What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa?

$\frac{9}{19}$

Got questions? Get instant answers now!

You could also do this problem by using a probability tree. See the Tree Diagrams (Optional) section of this chapter for examples.

Got questions? Get instant answers now!

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics' conversation and receive update notifications?

Ask

	1 Human A&P 2- Test 1 By Madison Christian Start Flashcards
	2 Human A&P 2- Test 2 By Madison Christian Start Flashcards
	2 Week 2 Social Psych By Yacoub Jayoghli Start Quiz
	Pre-class Online Quiz - Case Control Studies By Mucho Mizinduko Start Quiz
	2011 Dynamics CRM By Danielrosenberger Start Quiz
	English Proficiency Test By Anindyo Mukhopadhyay Start Quiz
	2 AP 02 Chemical Level of Organization Essay By OpenStax Start Flashcards
	5 AP 05 Integumentary System Essay By OpenStax Start Flashcards
	How well do you know 5SOS? By Wey Hey Start Quiz
	Worldport Outside By Rachel Carlisle Start Quiz