CHAPTER 5
Probability: What Are
the Chances?
5.2
Probability Rules
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Probability Rules
Learning Objectives
After this section, you should be able to:
 DESCRIBE a probability model for a chance process.
 USE basic probability rules, including the complement rule and the
addition rule for mutually exclusive events.
 USE a two-way table or Venn diagram to MODEL a chance process
and CALCULATE probabilities involving two events.
 USE the general addition rule to CALCULATE probabilities.
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Aim – How can we calculate probability values
from events?
H.W. – pg. 314 – 316 #43 – 48, 53, 55
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Probability Models
In Section 5.1, we used simulation to imitate chance behavior.
Fortunately, we don’t have to always rely on simulations to determine
the probability of a particular outcome.
Descriptions of chance behavior contain two parts:
The sample space S of a chance process is the set of all possible
outcomes.
A probability model is a description of some chance process that
consists of two parts:
• a sample space S and
• a probability for each outcome.
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Example: Building a probability model
Sample Space
36 Outcomes
The Practice of Statistics, 5th Edition
Since the dice are fair, each outcome is equally
likely.
Each outcome has probability 1/36.
5
Example – flipping a coin.
• Flip a coin 3 times and write down the probability model
for this:
• HHH, HHT, HTH, HTT, TTT, TTH, THT, THH
• BTW – your graphing calculator can simulate coin flips
and other experiments by doing the following:
• Going to apps
• Press 9
• Press 1
• Then simulate coin flipping
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Probability Models
Probability models allow us to find the probability of any collection of
outcomes.
An event is any collection of outcomes from some chance
process. That is, an event is a subset of the sample space. Events
are usually designated by capital letters, like A, B, C, and so on.
If A is any event, we write its probability as P(A).
In the dice-rolling example, suppose we define event A as “sum is 5.”
There are 4 outcomes that result in a sum of 5.
Since each outcome has probability 1/36, P(A) = 4/36.
Suppose event B is defined as “sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36
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Basic Rules of Probability
• The probability of any event is a number between 0 and 1.
• All possible outcomes together must have probabilities whose sum
is exactly 1.
• If all outcomes in the sample space are equally likely, the probability
that event A occurs can be found using the formula
P(A) =
number of outcomes corresponding to event A
total number of outcomes in sample space
• The probability that an event does not occur is 1 minus the
probability that the event does occur.
• If two events have no outcomes in common, the probability that one
or the other occurs is the sum of their individual probabilities.
Two events A and B are mutually exclusive (disjoint) if they
have no outcomes in common and so can never occur together—
that is, if P(A and B ) = 0.
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Basic Rules of Probability
We can summarize the basic probability rules more concisely in
symbolic form.
Basic Probability Rules
•For any event A, 0 ≤ P(A) ≤ 1.
•If S is the sample space in a probability model,
P(S) = 1.
•In the case of equally likely outcomes,
number of outcomes corresponding to event A
P(A) =
total number of outcomes in sample space
•Complement rule: P(AC) = 1 – P(A)
•Addition rule for mutually exclusive events: If A and B are
mutually exclusive,
P(A or B) = P(A) + P(B).
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A.P. Exam Common Error
• When using the complement rule, some students use
incorrect notation such as 0.25c = 0.75 or P(A)c = 0.75.
Remember that events have complements, not
probabilities. Here is a correct statement:
• If P(A) = 0.25, then P(Ac) = 1 – 0.25 = 0.75. Notice that
the c appears in the superscript of event A, not its
probability.
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Example and Practice:
• E-Book example: Distance learning.
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Practice:
• Randomly select a student who took the 2013 AP® Statistics exam
and record the student’s score. Here is the probability model:
• Show that this is a legitimate probability model
• Find the probability that the chosen student scored 3 or better.
Score:
Probability:
1
2
3
4
5
0.235
0.188
0.249
0.202
0.126
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Check our understanding:
• Choose an American adult at random. Define two
events:
• A = the person has a cholesterol of 240 milligrams per
deciliter of blood or above (high cholesterol).
• B = The person has a cholesterol of 200 to 239
milligrams per deciliter (borderline high cholesterol).
• According to the American Heart Association, P(A) =
0.16, P(B) = 0.29.
• Explain why events A and B are mutually exclusive.
• Solution:
• A person can’t have both high and borderline cholesterol.
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Check our understanding continued:
•
•
•
•
Find P(A or B).
Solution:
P(A or B) = P(A) + P(B) = 0.16 + 0.29 = 0.45.
If C is the event that the person chosen has normal cholesterol
(below 200 milligrams), what is P(C)?
• Solution:
• P(C) = 1 – 0.45 = 0.55
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Aim – How can we evaluate two way tables within
the context of probability measures?
• H.W. pg. 316 – 317 #49, 50, 53, 55bcd
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Two-Way Tables and Probability
When finding probabilities involving two events, a two-way table can
display the sample space in a way that makes probability calculations
easier.
Consider the example on page 309. Suppose we choose a student at
random. Find the probability that the student (a) has pierced ears.
(b) is a male with pierced ears.
(c) is a male or has pierced
ears.
Define events A: is male and B: has pierced ears.
(a)
(b) Each
(c)
We want
student
to find
is equally
P(male likely
or
and
pierced
pierced
to beears),
chosen.
ears),
that
that
103
is,is,
students
P(A
P(A
orand
B).have
There
B).
pierced
are
ears.
in
So,
theP(pierced
classofand
ears)
103
=
individuals
P(B)
103/178.
with
pierced
ears.There
Look90atmales
the intersection
the
“Male”
row=and
“Yes”
column.
are 19 males
with pierced
ears. So,
P(A
and B)
= 19/178.
However,
19 males
have pierced
ears
– don’t
count
them twice!
P(A or B) = (19 + 71 + 84)/178. So, P(A or B) = 174/178
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General Addition Rule for Two Events
We can’t use the addition rule for mutually exclusive events unless the
events have no outcomes in common.
General Addition Rule for Two Events
If A and B are any two events resulting from some chance
process, then
P(A or B) = P(A) + P(B) – P(A and B)
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Check understanding:
• Standard deck of playing cards (with jokers removed) consists of 52
cards in 4 suits – clubs, diamonds, spades, and hearts. Each suit
has 13 cards, with denominations of 2 – 10, jack, queen, king, and
ace. The face cards are the jack, queen, and king. If we shuffle the
deck thoroughly and deal one card. Define events as follows:
• F = getting a face card.
• H = getting a heart.
• Make a two-way table that displays the sample space.
• Find P(F and H).
• Explain why P(F or H) ≠ P(F) + P(H).
• Find P(F or H)
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Venn Diagrams and Probability
Because Venn diagrams have uses in other branches of mathematics,
some standard vocabulary and notation have been developed.
The complement AC contains exactly the outcomes that are not in A.
The events A and B are mutually exclusive (disjoint) because they do not
overlap. That is, they have no outcomes in common.
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Venn Diagrams and Probability
The intersection of events A and B (A ∩ B) is the set of all outcomes
in both events A and B.
The union of events A and B (A ∪ B) is the set of all outcomes in either
event A or B.
Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.
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Venn Diagrams and Probability
Recall the example on gender and pierced ears. We can use a Venn
diagram to display the information and determine probabilities.
Define events A: is male and B: has pierced ears.
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Practice – handout:
• Practice textbook and handout
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Probability Rules
Section Summary
In this section, we learned how to…
 DESCRIBE a probability model for a chance process.
 USE basic probability rules, including the complement rule and the
addition rule for mutually exclusive events.
 USE a two-way table or Venn diagram to MODEL a chance process
and CALCULATE probabilities involving two events.
 USE the general addition rule to CALCULATE probabilities.
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