Math Review
Lesson 10.3 – Probability and 2 Way Tables
Owen Kris
A two way table is a frequency table that displays data collected from one
source that belong to two different categories. Each entry in the table is called a joint
frequency, and the sums of the rows and columns are called marginal frequencies.
A joint relative frequency is the ratio of a frequency that is not in the total
row or the total column to the total number of values or observations. A marginal
relative frequency is the sum of the joint relative frequencies in a row or column. A
conditional relative frequency is the ratio of a joint relative frequency to the
marginal relative frequency. You can find a conditional relative frequency using a
row total or column total of a two-way table.
A question may ask you to determine whether a given data set is independent or
dependent. To solve this problem you must use an independence test. That means
that you use a formula that only works for an independent event, and if the equation
balances out, then they are independent, and if not they are dependent. The
formulas you can use for an independence test are P(A) x P(B) = P(A+B) or P(A/B) =
P(A). You just plug in the values for your particular data set and simplify to see if the
two sides of the equation are equal.
Some questions that might be asked are to fill in the two-way table with a few given
values, to convert it to a relative frequency table, to find the probability of
something in the table happening, or to test for independence.
Preparation
Studied
Pass
Grade
Total
6
Fail
Total
Did not study
10
38
50
Example problems:
1)
Fill in the two-way table. (Hint: remember that the marginal frequencies for
each row or column must add up to the total, and the joint frequencies in each row
or column must add up to the marginal frequencies.)
2)
You survey 171 males and180 females at Grand Central Station in New York
City. Of those, 132 males and 151 females wash their hands after using the public
restrooms. Organize these results in a two-way table. Then find and interpret the
marginal frequencies.
3)
Use the survey results from the last problem to make a two-way table with
joint and marginal relative frequencies.
4)
What is the probability that a randomly selected person in Grand Central
Station washes his or her hands after using the public restroom? What is the
probability that a randomly selected person is a man given that they do not wash
their hands after using the restroom?
5)
A survey finds that 110 people ate breakfast and 30 people skipped
breakfast. Of those who ate breakfast, 10 people felt tired. Of those who skipped
breakfast, 10 people felt tired. Make a two-way table that shows the conditional
relative frequencies based on the breakfast totals.
6)
Are eating breakfast and being tired independent events? Explain.
7)
Three different local hospitals in New York surveyed their patients. The
survey asked whether the patient’s physician communicated efficiently. The results,
given as joint relative frequencies, are shown in the two-way table.
Location
Response
Glens Falls
Saratoga
Albany
Yes
0.123
0.288
0.338
No
0.042
0.077
0.131
a.
What is the probability that a randomly selected patient located in Saratoga
was satisfied with the communication of the physician?
b.
What is the probability that a randomly selected patient who was not
satisfied with the physician’s communication is located in Glens Falls?
c.
Determine whether being satisfied with the communication of the physician
and living in Saratoga are independent events.
8)
You want to find the quickest route to school. You map out three routes.
Before school, you randomly select a route and record whether you are late or on
time. The table shows your findings. Assuming you leave at the same time each
morning, which route should you take? Explain.
On time
Late
Route A
Route B
Route C
Challenge:
The two-way table shows the number of people who are either right handed
or left handed as compared to gender. Use this table to determine the probability
that a randomly selected person is female given that she is right-handed.
Left
Female
Right
1/21
Total
115
Male
Total
196
231
Is there another way to do this? Show your work. (Hint: Use a formula that we have
worked with.)
Answer Key:
1)
Fill in the two-way table. (Hint: remember that the marginal frequencies for
each row or column must add up to the total, and the joint frequencies in each row
or column must add up to the marginal frequencies.)
Preparation
Grade
Studied
Did not study
Total
Pass
34
6
40
Fail
4
6
10
Total
38
12
50
2)
You survey 171 males and180 females at Grand Central Station in New York
City. Of those, 132 males and 151 females wash their hands after using the public
restrooms. Organize these results in a two-way table. Then find and interpret the
marginal frequencies.
Male
Female
Total
Washes Hands
132
151
283
Doesn't wash
hands
39
29
68
Total
171
180
351
The marginal frequencies are 283 and 68, and this means that 283 people wash
their hands while 68 people do not.
3)
Use the survey results from the last problem to make a two-way table with
joint and marginal relative frequencies.
Male
Female
Total
Washes hands
132/351
151/351
283/351
Doesn't wash
hands
39/351
29/351
68/351
Total
171/351
180/351
1
4)
What is the probability that a randomly selected person in Grand Central
Station washes his or her hands after using the public restroom? What is the
probability that a randomly selected person is a man given that they do not wash
their hands after using the restroom?
283/351 or 0.81
39/68 or 0.57
5)
A survey finds that 110 people ate breakfast and 30 people skipped
breakfast. Of those who ate breakfast, 10 people felt tired. Of those who skipped
breakfast, 10 people felt tired. Make a two-way table that shows the conditional
relative frequencies based on the breakfast totals.
Ate breakfast
Skipped breakfast
Total
Felt Tired
10/110
10/30
20/140
Did not feel tired
100/110
20/30
120/140
Total
1
1
1
6)
Are eating breakfast and being tired independent events? Explain.
P(A) x P(B) = P(A+B)
P(Ate breakfast) x P(felt tired) = P(Ate breakfast and felt tired)
(110/140) x (20/140) = 10/140
.11 = .07
They are dependent, because they fail the independence test.
7)
Three different local hospitals in New York surveyed their patients. The
survey asked whether the patient’s physician communicated efficiently. The results,
given as joint relative frequencies, are shown in the two-way table.
Location
Response
Glens Falls
Saratoga
Albany
Yes
0.123
0.288
0.338
No
0.042
0.077
0.131
a.
What is the probability that a randomly selected patient located in Saratoga
was satisfied with the communication of the physician?
.288 + .077 = .365
.288/.365 = 0.79
b.
What is the probability that a randomly selected patient who was not
satisfied with the physician’s communication is located in Glens Falls?
.123 + .288 + .338 = .749
.123/.749 = .16
c.
Determine whether being satisfied with the communication of the physician
and living in Saratoga are independent events.
P(A) x P(B) = P(A+B)
P(Satisfied) x P(Saratoga) = P(Satisfied in Saratoga)
.749 x .365 = .288
.273 = .288
They are not independent, because they fail the independence test.
8)
You want to find the quickest route to school. You map out three routes.
Before school, you randomly select a route and record whether you are late or on
time. The table shows your findings. Assuming you leave at the same time each
morning, which route should you take? Explain.
On time
Late
Route A
Route B
Route C
Route A = 7/11 = 0.64
Route B = 11/14 = 0.79
Route C = 12/16 = 0.75
Route B, because there is the highest percentage chance that you will get to school
on time.
Challenge:
The two-way table shows the number of people who are either right handed
or left handed as compared to gender. Use this table to determine the probability
that a randomly selected person is female given that she is right-handed.
Left
Right
Total
Female
1/21 = 11/231
104
115
Male
24
92
116
Total
35
196
231
104/196 or .53
Is there another way to do this? Show your work. (Hint: Use a formula that we have
worked with.)
P(A/B) = (P(B/A) x P(A))/P(B)
P(Female/Right handed) = (P(Right handed/female) x P(Female))/P(right handed)
P(A/B) = (104/115) x (115/231)/(196/231)
P(A/B) = 0.53