Section 4.1: Sample Spaces and Probability
Definition: Probability is the chance of an event occurring. A probability experiment is a chance process
that leads to well-defined results, called outcomes.
What is the outcome of a probability experiment? What is the sample space?
Example: Suppose we flip two fair coins. What is the sample space?
Example: Using a tree diagram, find the sample space for flipping three coins.
Definition: An event consists of the set of outcomes of a probability experiment. An event with one outcome
is called a simple event.
Example: Consider flipping three coins. Give an example of a simple event. Give an example of a compound
event.
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There are three different interpretations of probability. They are:
1. Classical
2. Empirical
3. Subjective
Classical Probability
In classical probability, we define
Probability of an event E =
number of outcomes in E
total number of outcomes
or
P (E) =
n(E)
.
n(S)
Example: What is the probability of getting exactly one tails when three coins are flipped?
Example: An card is drawn from an ordinary deck of cards. Find the probability of:
(a) drawing an 8
(b) drawing a 6 and a diamond
(c) drawing a 3 or a club (example of inclusive or)
(d) drawing a 3 or a 6 (example of exclusive or)
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Probability Rules:
1. 0 ≤ P (E) ≤ 1
2. if an event E cannot occur, P (E) = 0.
3. if an event E is certain to occur, P (E) = 1.
4. The sum of all probabilities of all outcomes in a sample space is 1.
Definition: The complement of an event E is the set of all outcomes in the sample space not included in
the outcomes of event E, and is denoted E.
Example: Find the complement of the event of rolling a die and getting a 3.
Note that P (E) + P (E) = 1. Sometimes it is easier to find the probability of a complement of an event and
then subtracting that probability from 1 to find the probability of the event.
Example: Two fair coins are tossed. Find the probability of obtaining at least one head.
Empirical Probability
Given a frequency distribution, the probability of an event being in a given class is
P (E) =
frequency for the class
f
= .
total frequency of the distribution
n
This probability is called the empirical probability and is based on observation.
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Example: In our questionnaire, when asked how one arrived at school, the following responses were recorded.
Mode of Transport
Frequency
Bus
1
Car
26
Walk
46
(a) Find the probability that a student walked to school.
(b) Find the probability that a student took a bus or car to school.
Law of Large Numbers states that...
Sections 4.2 and 4.3: Rules of Probability
Definition: Two events are mutually exclusive if the events cannot occur at the same time.
Example: Consider an ordinary deck of 52 playing cards. Give an example of two mutually exclusive events.
Give an example of two events which are not mutually exclusive. Let’s compute the probabilities that these
events will occur.
When two events are A and B are NOT mutually exclusive, then the probability that A OR B will occur is
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
If A and B are mutually exclusive events, then P (A ∩ B) = 0. We thus have the simplification
P (A ∪ B) = P (A) + P (B).
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Warning: We can only use P (A ∪ B) = P (A) + P (B) when A and B are mutually exclusive!!
Let’s examine the possibilities for events A and B using a Venn diagram.
Example: Using our questionnaire data, we’ll construct a contingency table for gender and smoking. The result
is as follows:
No
Yes
Missing
All
Female
36
5
1
41
Male
31
3
0
34
Missing
1
0
0
*
All
67
8
*
75
(a) Find the probability that one of my students is a female smoker.
(b) Find the probability that one of my students is a female or a smoker.
Definition: Two events A and B are independent events if the fact that A occurs does NOT affect the
probability of B occurring.
Example: Are the events flipping a fair coin and getting heads and rolling a fair die and getting a 2 independent?
Are the events picking a spade from an ordinary deck of playing cards and then, without replacement, picking
a heart from the deck independent?
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When two events A and B are independent, the probability of both A and B occurring is
P (A ∩ B) = P (A) · P (B).
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Example: Find the probability of flipping a fair coin and getting heads and rolling a fair die and getting a 2.
Example: A card is drawn from a deck and replaced. Find the probability of getting a spade and then a heart.
Definition: We say that two events are dependent if the occurrence of the first event changes the probability
of the occurrence of the second event.
Definition: The conditional probability of the event B given the event A is the probability that event B
occurs after event A has already occurred. The notation is P (B|A).
When two events are dependent, the probability of both occurring is
P (A ∩ B) = P (A) · P (B|A).
Example: A card is drawn from a deck and NOT replaced. Find the probability of getting a spade and then
a heart.
For events A and B, we know that P (A ∩ B) = P (A) · P (B|A). Solving for P (B|A) gives us:
Example: Consider again the smoking and gender contingency table in the earlier example.
(a) Find the probability that a student smokes given that she is female.
(b) Find the probability that a student is female given that the student smokes.
(c) Are the events that a student smokes and that a student is female independent?
Example: A fair coin is tossed three times. Find the probability of getting at least one head.
Sections 4.4 and 4.5: Counting Rules and Probability
Example: Suppose that I have 5 shirts, 4 pairs of pants, and 3 sport coats. Assuming I don’t care about
matching, how many possible outfits do I have?
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Fundamental Counting Rule: In a sequence of events in which the first one has k1 possibilities and the
second event has k2 possibilities, and the third has k3 , and so forth, the total number of possibilities of the
sequence will be
k1 · k2 · k3 · · · kn .
Example: Toll collectors are identified by four-digit numbers using the digits 0 through 9.
(a) How many different numbers are possible if repetition of digits is allowed?
(b) How many different numbers are possible if repetition of digits is not allowed?
Notation: We define n! = n(n − 1)(n − 2) · · · 2 · 1 and define 0! = 1.
Definition: A permutation is an arrangement of n objects in a specific order.
Permutation Rule: The arrangement of n objects in a specific order using r objects at a time is called a
permutation of n objects taking r objects at a time. It is written n Pr and the formula is
n Pr
=
n!
.
(n − r)!
Example: Compute 5 P3 .
Example: How many different ways can a president, vice-president, and secretary be chosen for student council
among the 10 members of the council?
Definition: A selection of distinct objects without regard to order is called a combination.
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Example: The student council of 10 members would like to form a committee consisting of four members to
look at improving school lunches. How many possible committees are there?
Combination Rule: The number of combinations of r objects selected from n objects is denoted n Cr and is
given by the formula
n Cr
=
n!
.
(n − r)!r!
Example: Compute 5 C3 .
Note: Combinations are used when the order or arrangement is not important.
Example: Our student council consists of 6 girls and 4 boys. If the committee for school lunches must consist
of 2 girls and 2 boys, how many different committees are possible?
Example: In a train yard, there are 4 tank cars, 12 boxcars, and 7 flatcars. How many ways can a train be
made up consisting of 3 tank cars, 4 boxcars, and 2 flatcars?
Example: (a) Find the number of possible hands in five-card poker.
(b) Find the number of possible five card hands that are all spades.
(c) Find the probability of an all-spade five-card hand.
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Example: A box contains 10 resistors, four of which are defective. If four are sold at random, find the following
probabilities:
(a) exactly two are defective
(b) none is defective
(c) all are defective
(d) at least one is defective
Example: Consider our student council consisting of 6 girls and 4 boys. Three are selected to meet with the
superintendent. Find these probabilities:
(a) all three selected will be girls
(b) all three selected will be boys
(c) two girls and one boy will be selected
(d) one girl and two boys will be selected
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