Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
Conditional Probability—Part 2
Objective: You will use tables to calculate the probability that an event happens given that another event
happened.
Class Problem: disease testing
Consider the situation of a medical test for a disease.
Most often, the test accurately indicates that a person has the disease (“positive”) or does not have the
disease (“negative”). But the test is not perfect, so sometimes the test comes out positive even though the
person doesn’t have the disease (a “false positive”) and sometimes the test comes out negative even
though the person does have the disease (a “false negative”).
Question: suppose a person receives a positive test for the
disease. What is the probability that he or she actually has the
disease? To answer this question, we will make a table.
Consider the following situation: In a population being tested for a disease, 3% have the disease
and 97% do not. Also, when someone with the disease is tested, there’s a 98% chance that the test
result is positive. When someone not having the disease is tested, there’s a 5% chance that the test
result is positive. Now, consider 10,000 people who have been tested for this disease.
A. Fill in the following table. (Hint: fill in the shaded part first):
Number of people
who test Positive
Number of people
who test Negative
Total number of
people
Number of people
who have the disease
Number of people
who do not have the
disease
Total number of
people
10,000
Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
B. Check your table:
Number of people
who test Positive
Number of people
who test Negative
Total number of
people
Number of people
who have the disease
294
6
300
Number of people
who do not have the
disease
Total number of
people
485
9215
9700
779
9221
10,000
C. Find the probability of accurate positive result, or what is the probability that a person has the disease
AND tests positive:
P(disease and positive) =
# of people who have the disease and test positive
294
= 0.0294

total number of people tested
10,000
D. What is the probability that the test gives a “false positive” (no disease, but the test outcome is
positive)?
Remarkably, false positive results happen more often than accurate positive tests! (This happens on many
real tests for diseases, and many people are unaware of this fact.)
E. What is the probability that the test result is positive? (This would include both the accurate and the
false positives)
F. What is the probability that the test gives an accurate negative result?
G. What is the probability that the test gives a “false negative”?
Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
H. What is the probability that the test result is negative?
I. What is the probability that the test gives an accurate result (either positive or negative)?
J. What is the probability that the test gives an inaccurate result (either a “false positive” or a “false
negative”)?
Another type of question:
Now here’s the new type of question we will learn to answer today: What is the probability that someone
who tests positive actually has the disease? And also: What is the probability that someone who tests
positive does not have the disease?
Here is the calculation using the conditional probability notation:
P(disease | positive) 
# of people who have the disease and test positive 294

 0.377
# of people who test positive
779
P(no disease | positive) 
# of people who don' t have the disease and test positive 485

 0.623
# of people who test positive
779
So in this problem, a person who tests positive is almost twice as likely not to have the disease as to have
it!
Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
Homework Problems:
1. Suppose that there’s another test for the disease that has better accuracy. As before, suppose that 3%
have the disease and 97% do not. When someone with the disease is tested, there’s a 99% chance that
the test result is positive. When someone not having the disease is tested, there’s a 99.2% chance that
the test result is negative.
a. Fill in the table with the number in each category in a sample of 10,000 people. Don’t round the
fractional people.
Number of people
who test Positive
Number of people
who test Negative
Total number of
people
Number of people
who have the disease
Does not have the
disease
Total number of
people
10,000
b. What is the probability that the test gives an accurate positive result?
c. What is the probability that the test result is positive?
d. What is the probability that someone who tests positive actually has the disease?
e. What is the probability that someone who tests negative actually has the disease?
Name:_________________________________
Algebra 2
June 2 (A Block) June 3 (E Block), 2014
Conditional probability using tables
2. During a flu epidemic, 35% of a city’s 10,000 students have the flu. Of those with the flu, 90% have
high temperatures. However, a high temperature is also possible for people without the flu. The
school nurse estimates that 12% of those without the flu have high temperatures too.
a. Fill in the table.
(Hint: Think of “high temperature” as being a test for the flu.)
Total number of
people
Total number of
people
10,000
b. What percent of students have a high temperature?
c. If a student has a high temperature, what is the probability that the student has the flu?
Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
3. A survey is taken of how a sample of 1000 people like to listen to music. Of those surveyed,
12.5% were under 25 years old, 50% were age 25-40, and 37.5% were over 40.
The results were as follows:
Under 25:
20% usually listen to the radio, 80% usually listen to Pandora
Age 25-40: 55% usually listen to the radio, 45% usually listen to Pandora
Over 40:
80% usually listen to the radio, 20% usually listen to Pandora
a. Make a table that represents the data
b. If a person is randomly selected from the sample and found to listen to Pandora, what is the
probability that he or she is under 25?
c. If a person is randomly selected from the sample and found to listen mostly to the radio,
what is the probability that he or she is over 40?
Name:_________________________________
June 2 (A Block) June 3 (E Block), 2014
Algebra 2
Conditional probability using tables
Answers:
1. a.
Number of people
who test Positive
Number of people
who test Negative
Total number of
people
Number of people
who have the disease
297
3
300
Does not have the
disease
77.6
9622.4
9700
Total number of
people
374.6
9625.4
10,000
b. 0.0297
c. 0.03746
d. 0.7928
e.
3
 0.00031
9625.4
2. a.
High Temperature
Normal Temperature
Total number of
people
Have the flu
3150
350
3500
No flu
7780
5720
6500
Total number of
people
3930
6070
10,000
b. 39.3% of students have a high temperature.
c.
3150
 0.8015 , or about 80%.
3930
3. a.
b.

Listens to radio
Listens to Pandora
Totals
Under 25
25
100
125
25-40
275
225
500
Over 40
300
75
375
Totals
600
400
10,000
100
 25%
400

c.
300
 50%
600