Math 243
Sections 3.2-3.5
Probability Rules!!
Overview
• Venn Diagrams
• General Addition Rule
• General Multiplication Rule
• Contingency Tables
• Conditional Probability Rule
• Independent Event Test
• Probability Trees
Example 1. Suppose that of the 34 students in a stats class, 82% like chocolate, 59% like espresso,
and 53% like both chocolate and espresso.
a. Define the events and draw a Venn Diagram to represent them.
b. What’s the probability that a randomly chosen student likes chocolate but not espresso?
c. What’s the probability that a randomly chosen student likes espresso but not chocolate?
d. What’s the probability that a randomly chosen student likes neither espresso nor chocolate?
e. What’s the probability that a randomly chosen student likes chocolate or espresso?
General Addition Rule: P(A or B) = P(A) + P(B) − P(A and B) for events A and B.
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Disjoint Events (Mutually Exclusive)
P(A or B) = P(A) + P(B)
Events that are not disjoint
P(A or B) = P(A) + P(B) − P(A and B)
Example 2. Suppose that for a given term, 20% of PCC students take courses that meet in the
evening, 31% take online courses, and 8% take both evening courses and online courses.
a. Are the events “take evening courses” and “take online courses” disjoint? Why or why not?
b. Draw a Venn Diagram.
c. What is the probability that a randomly selected student takes an online course and not an evening
course?
d. What’s the probability that a randomly selected student takes either an online course or an evening
course?
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Example 3. We’re going to continue our chocolate/espresso example. (Where 82% like chocolate,
59% like espresso, and 53% like both chocolate and espresso). This time, we’ll represent the
probabilities of these events using a Contingency Table. The given numbers are already in the table
and we can use them to compute the rest.
Table 1. Chocolate and Espresso Likability
Likes
Chocolate
Likes
Espresso
Does not like
chocolate
0.53
Total
0.59
Does not like
espresso
Total
0.82
a. What’s the probability that a randomly selected student likes espresso given that they like
chocolate? Find using the table.
b. What percentage of students who like chocolate like espresso? Find using the conditional
probability rule below.
c. What’s the probability that a randomly selected student does not like chocolate given that they like
espresso? Find using the table.
d. What percentage of students who like espresso do not like chocolate? Find using the conditional
probability rule.
Conditional Probability Rule: 𝑷𝑷(𝑩𝑩|𝑨𝑨) =
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𝑷𝑷(𝑩𝑩 𝒂𝒂𝒂𝒂𝒂𝒂 𝑨𝑨)
𝑷𝑷(𝑨𝑨)
for events A and B.
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Independent Events Test: Events A and B are independent if P(B|A)=P(B). If this is true
then knowing that A occurred does not change the chance of B occurring.
Example 4. Continuing the chocolate/espresso example, do you think liking chocolate is independent
of liking espresso (for this group of students)?
To test independence, find P(E|C) and P(E) and see if they are equal.
Example 5. What does independence look like in a contingency table? Here are the results from a
Psychology class where 80% like Chocolate, 40% like Espresso and 32% like both.
Table 2. Chocolate and Espresso Likability
Likes
Chocolate
Likes
Espresso
Does not like
chocolate
0.32
Total
0.40
Does not like
espresso
Total
0.80
Are the events liking chocolate and liking espresso independent for this class?
Conditional Probability Rule: 𝑷𝑷(𝑩𝑩|𝑨𝑨) =
𝑷𝑷(𝑩𝑩 𝒂𝒂𝒂𝒂𝒂𝒂 𝑨𝑨)
𝑷𝑷(𝑨𝑨)
for events A and B.
An alternate way that this formula can be used is:
General Multiplication Rule: 𝑷𝑷(𝑩𝑩 𝒂𝒂𝒂𝒂𝒂𝒂 𝑨𝑨) = 𝑷𝑷(𝑩𝑩|𝑨𝑨) · 𝑷𝑷(𝑨𝑨)
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Tree Diagrams can be used to illustrate sample spaces provided that the alternatives are not too
numerous. Once the sample space is illustrated a tree diagram can be used for determining
probabilities. Use them when you have successive events or conditional probabilities.
Example 6: Julio is flying from Boston to Denver with a connection in Chicago. The probability that
his first flight leaves on time is 0.15. If the flight is on time, the probability that his luggage will make
the connecting flight in Chicago is 0.95, but if the first flight is delayed, the probability that the
luggage will make it is only 0.65.
(a) Draw a tree diagram of the events.
(b) What is the probability that his luggage arrives in Denver with him?
(c) Given that his luggage arrived with him, what is the probability that his first flight left on time?
Note that the conditional probability has been reversed. There is a formula called Bayes Theorem but
it is easier to calculate it using the formula and the tree.
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Practice 1. Suppose that employment data for a large company shows that 72% of workers are
married, 46% are college graduates, and 22% are both married and college graduates. Create a Venn
diagram displaying these events.
What is the probability that a worker
(a) is neither married nor a college graduate?
(b) is married but not a college graduate?
Practice 2. How are the smoking habits of students related to their parents’ smoking? Here is a
contingency table of data from a survey of students in 8 Oregon high schools.
Both
parents
smoke
One
Parent
Smokes
Neither
parent
smokes
Student
Smokes
400
416
188
Student does
not smoke
1380
1823
1168
Total
Total
Find the following probabilities.
(a) P ( student smokes )
(b) P ( neither parent smokes )
(c) P ( at least 1 parent smokes ) ?
(d) P ( student smokes and 1 parent smokes ) ?
(e) P ( student smokes and neither parent smokes )
(f) P ( student smokes & at least 1 parent smokes )
(g) What is the probability that a student who smokes has neither parent that smokes?
(h) What is the probability that if both parents smoke, their child will smoke?
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Practice 3. A 2005 study published a report on the reliability of HIV testing. Results of this study
suggested that among people with HIV, 99.7% of tests conducted were (correctly) positive, while for
people without HIV 98.5% of tests were (correctly) negative. Assume that a clinic administering HIV
testing has 15% of patients that actually carry HIV.
(a) Create a Tree Diagram to display these events.
(b) What is the probability that someone carries HIV and tests negative?
(c) What is the probability that someone tests negative?
(d) What is the probability that someone does not carry HIV given that they tested positive?
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