4.4 Contingency tables

Collaborative statistics for mt 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

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

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

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

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

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

$\frac{280}{755}$

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

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

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

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

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

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

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

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.

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.

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 $P(F OR P) = \frac{45}{100} + \frac{25}{100} - \frac{11}{100} = \frac{59}{100}$

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

What is the probability that Alissa does not catch Muddy?

$\frac{41}{60}$

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

$\frac{9}{19}$

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

Questions & Answers

it is the relatively stable flow of income

Chidubem Reply

what is circular flow of income

Divine Reply

branches of macroeconomics

SHEDRACK Reply

what is Flexible exchang rate?

poudel Reply

is gdp a reliable measurement of wealth

Atega Reply

introduction to econometrics

Husseini Reply

mostafa

LEMLEM

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Sammol

Mahesh

Ruqayat

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Nyawa

hey

Sammol

God

hello

Jahara

Good morning

Jorge

abubakar

Nmesoma

Mahesh

Tom

Why is unemployment rate never zero at full employment?

Priyanka Reply

bcoz of existence of frictional unemployment in our economy.

Umashankar

what is flexible exchang rate?

poudel

due to existence of the pple with disabilities

Abdulraufu

the demand of a good rises, causing the demand for another good to fall

Rushawn Reply

is it possible to leave every good at the same level

Joseph

I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish

Anuolu

is not really possible to let the value of a goods to be same at the same time.....

Salome

Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.

Geetha Reply

Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.

Waeth

discus major problems of macroeconomics

Alii Reply

what is the problem of macroeconomics

Yoal

Economic growth Stable prices and low unemployment

Ephraim

explain inflationcause and itis degre

Miresa Reply

what is inflation

Getu

increase in general price levels

WEETO

Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?

Chisomo Reply

how to calculate actual output?

Chisomo

how to calculate the equilibrium income

Beshir

Criteria for determining money supply

Thapase Reply

who we can define macroeconomics in one line

Muhammad

Aggregate demand

Mohammed

C=k100 +9y and i=k50.calculate the equilibrium level of output

Mercy Reply

Isiaka

Geli

Man

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Bahunda

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nd u

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Sekou

ma management ho

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Amisha

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Amisha

belatari

Amisha

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nd u

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ahh

Amisha

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Amisha

money as unit of account means what?

Kalombe

A unit of account is something that can be used to value goods and services and make calculations

Jim

all of you please speak in English I can't understand you're language

Muhammad

I want to know how can we define macroeconomics in one line

Muhammad

it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500

Kalombe

Mercy is it clear?😋

Kalombe

hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.

Galge Reply

if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️

Kalombe

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Source: OpenStax, Collaborative statistics for mt230. OpenStax CNX. Aug 18, 2011 Download for free at http://legacy.cnx.org/content/col11345/1.2

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