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Probability Notes
3.1
Probability—the likelihood that an event will occur.
Consider the following example: During class, Dr. Rivera randomly selected a student
and asked whether the student’s major was early childhood or middle grades.
Ask yourself the following questions:
1. How many answers did the student give? (one)
2. What were the possible answers? (early childhood or middle grades)
When a process results in just one answer, we call it an experiment.
Experiment—a process that when performed results in one and only one of many
observations.
Outcomes—the observations of the experiment.
Sample Space—the collection of all outcomes for an experiment; denoted by S.
From the example,
 Experiment: selecting a student and asking major
 Outcomes: early childhood, middle grades
 Sample space: S = {early childhood, middle grades}
Venn diagrams and tree diagrams are often used to represent the sample space of an
experiment. The Venn diagram and tree diagram for this example are below.
S
* early childhood
*middle grades
Early
childhood
S
Middle
grades
Suppose now that the experiment is changed to asking two students for their majors.
What are the possible outcomes of this experiment? To make this easier, let’s abbreviate
early childhood as EC and middle grades as MG. The outcomes are EC EC, EC MG,
MG EC, and MG MG.
Event—a collection of one or more of the outcomes of the experiment.
Example: For this new experiment an event might be that both students selected were
middle grades majors. Another event might be that at least one of the students was an
early childhood major.
Simple event—an event that includes one and only one of the final outcomes for an
experiment.
Example: The event that both students selected are middle grades majors is a simple
event. Of all of the possible outcomes (i.e. EC EC, EC MG, MG EC, MG MG) the only
outcome described by this simple event is MG MG.
Compound event—a collection of more than one outcome for an experiment.
Example: The event that at least one of the students is an early childhood major is a
compound event. Of all of the possible outcomes (i.e. EC EC, EC MG, MG EC, MG
MG), the outcomes EC EC, EC MG, MG EC are described by this compound event.
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3.2
A Random Event is one in which we are not sure of exactly what will happen even
though we know the possible outcomes and we can predict a long term pattern.
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3.3/3.4
Probability—a numerical measure of the likelihood that a specific event will occur; if
the event is A, then the probability that A will occur is denoted P(A).
Example: Flip a coin. What is the probability of heads? This is denoted P(heads).
Properties of Probability
1. The probability of an event always lies in the range of zero to one.
Impossible event—an event that absolutely cannot occur; probability is zero.
Example: Suppose you roll a normal die. What is the probability that you will get
a seven? P(7) = 0.
Sure event—an event that is certain to occur; probability is one.
Example: Suppose you roll a normal die. What is the probability that you will get
a number less than 7? P(a number less than 7) = 1.
2. The sum of the probabilities of all simple events (or final outcomes) for an
experiment is always one.
Example: Suppose you flip two coins. What are the outcomes? HH, HT, TH, TT
This rule says that the probabilities of each of these outcomes should sum to one.
That is,
P(HH) + P(HT) + P(TH) + P(TT) = 1
Two Types of Probability
1. Experimental Probability—probability estimates arrived at using data gathered
through experiments.
2. Theoretical Probability—the specific value that is approached by experimental
probability as more experiments are run.
Three Approaches to Calculating Probability
1. Classical Probability
o In this approach, outcomes are assumed to be equally likely.
o Equally likely outcomes—two or more outcomes (or events) that have the
same probability of occurrence.
o Example: When you flip a fair coin, the probability of getting heads is
equal to the probability of getting tails. Heads and Tails are equally likely
outcomes.
o If the outcomes are equally likely, then we use the classical probability
rule to calculate the probability.
o Classical Probability Rule—P(A) = (# of successes)/(Total # of
outcomes)
o Example: Suppose you roll a die.
 P(5) = 1/6
 P(odd) = 3/6 = 1/2
 P(not 4) = 5/6
2. Relative Frequency
o If the outcomes for an experiment are not equally likely, we cannot use the
Classical Probability Rule.
o Instead, we perform the experiment again and again to generate data. We
use the relative frequencies from this data to approximate the probability.
o Example: What is the probability that the next car that comes out of an
auto factory is a “lemon?” In this experiment, the outcomes are “lemon”
and “not lemon.” Clearly, these two outcomes are not equally likely. If
they were, then half of the cars produced would be a lemon. So, to find
the probability of a lemon we must conduct an experiment repeatedly and
gather some data. Suppose we go to the auto plant and check the next 500
cars to see if they are lemons. We find that 10 out of the 500 are lemons.
Using this information, we can approximate the probability that a car is a
lemon using the relative frequency.
P(lemon) = 10/500 = .02
o Relative Frequency as an Approximation of Probability—n = # of
times you repeat the experiment; f = # of times an event A is observed;
P(A) = f/n.
o From the example, the probability that we computed is not exact. It is
only an approximation. How could we get a better approximation? Check
a larger number or cars.
o Law of Large Numbers—if an experiment is repeated again and again,
the probability of an event obtained from the relative frequency
approaches the actual or theoretical probability.
3. Subjective Probability
o Consider this question: What is the probability that Auburn will beat
Georgia this year? The question does not represent equally likely
outcomes. Nor does it represent an experiment that can be conducted over
and over again. When this happens, we have to use subjective probability.
o Subjective probability—the probability assigned to an event based on
subjective judgment, experience, information, and belief.
o For the above question, Dr. Rivera may say the probability is .95 or 95%
whereas someone else (a Georgia fan perhaps) may say the probability is
0, i.e. that it is not going to happen.
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Marginal and Conditional Probabilities
Suppose the faculty at a local school were polled as to their agreement/disagreement with
the following statement: Coaches should be paid more than regular classroom teachers.
The following two-way table contains the results.
AGREE DISAGREE
MALE
20
10
FEMALE
15
35
From such a table, we can compute two types of probability—marginal and conditional.
First, you should add a row and column to the table for totals.
AGREE DISAGREE TOTAL
MALE
20
10
30
FEMALE
15
35
50
TOTAL
35
45
80
Marginal Probability—the probability of a single event without consideration of any
other event; also called simple probability.
Example: P(male) = (# of males)/(total # of teachers) = 30/80
Example: P(agree) = (# of teachers who agree)/(total # of teachers) = 35/80
Conditional Probability—the probability that an event will occur given that another
event has already occurred.
Example: Suppose that one teacher is selected at random. It is known that the teacher is a
male. What is the probability that the teacher agrees?
P(agrees male)  (# of males who agree) /(total # of males)  20 / 30 .
P(agrees male) is read “the probability the teacher agrees given that the teacher is a
male.”
Example: P(male agrees)  (# of males who agree) /(total # who agree)  20 / 35 .
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3.5
Geometric Probability


In some cases it is not possible to compute the theoretical probability of an event
by counting successes over total.
Geometric probability is another approach to computing probability.
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Fair Games, Odds, & Counting
Odds—the ratio of the probability that an event will occur to the probability that the
event will not occur.
Example: Flip a coin twice. What are the odds of getting at least one head?
Solution: We need the ratio P(at least one head)/P(no heads). That is (3/4)(1/4) which is
3:1.
Rule: Given the odds of an event occurring is m:n, the probability of that even occurring
is m/(m + n).
Example: The odds of winning a game are 1:45. What is the probability of winning?
Solution: P(winning) = 1/(45 + 1) = 1/46.
The Counting Rule—if an experiment consists of three steps and if the first step can
result in m outcomes, the second step in n outcomes, and the third step in k outcomes,
then the total number of outcomes for the experiment is m*n*k. This rule can be
extended to experiments with more or less steps.
Example: If a coin is flipped 3 times, how many outcomes are possible?
Solution: Each time the coin is flipped, there are 2 outcomes (heads or tails). 2*2*2 = 8
total outcomes.
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Mutually Exclusive Events & Independent/Dependent Events
Mutually Exclusive Events—events that cannot occur together.
Example: A card is chosen at random from a standard deck of 52 playing cards. Consider
the event “the card is a 5” and the event “the card is a king.” These two events are
mutually exclusive because there is no way for both events to occur at the same time.
Non-example: A card is chosen at random from a standard deck of 52 playing cards.
Consider the event “the card is a heart” and the event “the card is a king.” These two
events are not mutually exclusive because it is possible for these two events to occur at
the same time, namely when the King of Hearts is selected.
Independent Events—two events are independent if the occurrence of one does not
affect the probability of the occurrence of the other; the way to check whether or not the
events are independent is to check to see if P( A B)  PA) or P( B A)  P( B).
Dependent Events—two events are dependent if the occurrence of one affects the
occurrence of the other.
Example: Consider the following two-way table.
MALE
FEMALE
TOTAL
YES NO TOTAL
15
45
60
4
36
40
19
81
100
Are the events “female” and “yes” independent?
Does P( female yes)  P( female) ?
P( female yes)  4 /19  0.211
P(female) = 40/100 = 0.4
These events are dependent. In terms of the problem, this means that the probability that
someone says yes depends on whether you are asking a male or a female.
Example: Consider the following two-way table.
DEFECTIVE GOOD TOTAL
MACHINE 1
9
51
60
MACHINE 2
6
34
40
TOTAL
15
85
100
Are the events “defective” and “Machine 1” independent?
Does P(defective machine 1)  P(defective) ?
P(defective machine1)  9 / 60  0.15
P(defective) = 15/100 = 0.15
These events are independent. In terms of the problem, this means that the probability
that a tape is defective does not depend on which machine produced the tape.
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5.2
Multiplication Rule for Independent Events—the probability of two independent
events A and B occurring together:
P(A and B) = P(A) * P(B)
Example: Let the experiment be flipping a coin and then rolling a die. What is the
probability of getting “heads” and “4?” First, are these independent events? Yes, they
are because the occurrence of one does not affect the probability of the occurrence of the
other. Therefore, we can use the multiplication rule from above.
P(heads and 4) = P(heads) * P(4) = (1/2) * (1/6) = 1/12
Multiplication Rule for Dependent Events—P(A and B) = P( A) * P( B A) .
Example: A bag contains 4 green balls and 3 red balls. If two balls are drawn from the
bag, what is the probability that both are green?
Events: A = 1st ball is green; B = 2nd ball is green
Are A and B independent or dependent events? Dependent
P(A and B) = P( A) * P( B A) = 4/7 * 3/6 = 12/42 or 0.2857
Using the Multiplication Rule for Dependent Events to compute Conditional Probability:
Example: The probability that a randomly selected student from a college is a senior is
0.2. The probability that a randomly selected student is a computer science major and a
senior is 0.03. Find the conditional probability that a randomly selected student is a
computer science major given that he/she is a senior.
Solution: Two events: S = senior and C = computer science major
We know P(S) = 0.2 and P(S and C) = 0.03. We are looking for P(C S ).
P(S and C) = P(S) * P(C S )
So, 0.03 = 0.2x and x = 0.15.
Joint Probability of Mutually Exclusive Events—if A and B are mutually exclusive
events, then P(A and B) = 0.
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5.3
Complementary Events—two mutually exclusive events that taken together include all
outcomes for an experiment.
Example: Let the experiment be rolling a die. Let A be the event that the number rolled
is odd. Then, the complement of A is the number rolled is even.
Example: Let the experiment be selecting a card from a standard deck of 52 cards. Let A
be the event the card is a heart. Then, the complement of A is the card is not a heart.
Note: P(A) + P(complement of A) = 1.
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Union of Events
Union of Events—the union of events A and B is the collection of all outcomes that
belong either to A or to B or to both A and B; it is denoted “A or B” or “ A  B ”
Probability of the Union of Two EventsP(A or B) = P(A) + P(B) – P(A and B)
Example: The following table gives a 2-way classification of all faculty members at a
local school.
TENURED NON-TENURED TOTAL
MALE
74
28
102
FEMALE
29
12
41
TOTAL
103
40
143
One faculty member is selected at random. Find each probability.


P(female or non-tenured)
P(female or non-tenured) = P(female) + P(non-tenured) – P(female and nontenured)
= (41/143) + (40/143) – (12/143)
= (69/143)
= 0.483
P(tenured or male)
P(tenured or male) = P(tenured) + P(male) – P(tenured and male)
= (103/143) + (102/143) – (74/143)
= (131/143)
= 0.916
Probability of the Union of Two Mutually Exclusive EventsP(A or B) = P(A) + P(B)
Example: The probability of a student getting an A in this course is 0.24 and that of
getting a B is 0.28. What is the probability that a randomly selected student from this
class will get an A or a B?
P(A or B) = P(A) + P(B)
= 0.24 + 0.28
= 0.52
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Combinations
Combinations
 The number of ways x elements can be selected from n elements.
 The order in which the elements are chosen does not matter.
 Denoted


n
 
 x
This is read “the number of combinations of n elements selected x at a time” or “n
choose x.”
The formula for counting combinations is (# of outcomes if order mattered)/(# of
ways to arrange x elements).
Example: Suppose that you wanted to choose 3 out of the first 5 letters of the alphabet.
How many combinations are possible?



If order mattered, how may outcomes would there be? 5*4*3 = 60 outcomes.
How many ways can you arrange a set of three letters? 3*2*1 = 6 ways.
Number of combinations possible: 60 divided by 6 = 10 combinations possible.
Just so that you will understand what this number is telling you, here are the ten
combinations of the first 5 letter taken 3 at a time.
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE