Probability
I
Introduction to Probability
A Satisfactory outcomes vs. total outcomes
B Basic Properties
C Terminology
II Combinatory Probability
A The Addition Rule – “Or”
1. The special addition rule (mutually exclusive events)
2. The general addition rule (non-mutually exclusive events)
B The Multiplication Rule – “And”
1. The special multiplication rule (for independent events)
2. The general multiplication rule (for non-independent events)
Probability for Equally Likely Outcomes
Suppose an experiment has N possible outcomes, all equally
likely. Then the probability that a specified event occurs equals
the number of ways, f, that the event can occur, divided by the
total number of possible outcomes. In symbols
Number of ways a given event can occur
f
Probability of a given event =
N
Total of all possible outcomes
Frequency distribution of annual income for U.S.
families
Probability from Frequency Distributions
What is the a priori probability
of having an income between
$15,000 and $24,999
Frequency distribution for students’ ages
N = 40
Frequency distribution for students’ ages
What is the likelihood of randomly selecting a student who is
older than 20 but less than 22?
What is the likelihood of selecting a student who’s age is an
odd number?
What is the likelihood of selecting a student who is either 21
or 23?
Sample space for rolling a die once
Possible outcomes for rolling a pair of dice
Probabilities of 2 throws of the die
• What is the probability of a 1 and a 3? 2/36
• What is the probability of two sixes? 1/36
• What is the probability of at least one 3? 12/36
The Sum of Two Die Tosses
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
What is the probability that the
sum will be
5?
7?
4/36
6/36
What is the probability that the
sum will be 10 or more? 6/36
What is the probability that the
sum will be either 3 or less or 11
or more? 3/36 + 3/36
Two computer simulations of tossing a balanced coin
100 times
Basic Properties of Probabilities
Property 1: The probability of an event is always between 0
and 1, inclusive.
Property 2: The probability of an event that cannot occur is 0.
(An event that cannot occur is called an impossible event.)
Property 3: The probability of an event that must occur is 1.
(An event that must occur is called a certain event.)
A deck of playing cards
The event the king of hearts is selected
1/52
The event a king is selected
1/13 = 4/52
The event a heart is selected
1/4 = 13/52
The event a face card is selected
3/13=13/52
Sample Space and Events
Sample space: The collection of all possible
outcomes for an experiment.
Event: A collection of outcomes for the
experiment, that is, any subset of the sample
space.
Probability Notation
If E is an event, then P(E) stands for the
probability that event E occurs. It is read “the
probability of E”
Venn diagram for event E
Relationships Among Events
(not E): The event that “E does not occur.”
(A & B): The event that “both A and B occur.”
(A or B): The event that “either A or B or both
occur.”
Event (not E) where E is the probability of drawing a
face card.
40/52=10/13
An event and its complement
The Complementation Rule
For any event E,
P(E) = 1 – P (~ E).
In words, the probability that an event occurs equals 1
minus the probability that it does not occur.
Combinations of Events
The Addition Rule – “Or”
• The special addition rule (mutually exclusive events)
• The general addition rule (non-mutually exclusive events)
The Multiplication Rule – “And”
• The special multiplication rule (for independent events)
• The general multiplication rule (for non-independent
events)
Venn diagrams for
(a) event (not E)
(b) event (A & B)
(c) event (A or B)
Event (B & C)
1/13 X 1/4 = 1/52
Event (B or C)
16/52 = 4/52 + 13/52-1/52
Event (C & D)
3/52 = 3/13 X 1/4
Mutually Exclusive Events
Two or more events are said to be mutually exclusive if at
most one of them can occur when the experiment is performed,
that is, if no two of them have outcomes in common
Two mutually exclusive events
(a) Two mutually exclusive events
(b) Two non-mutually exclusive events
(a) Three mutually exclusive events (b) Three nonmutually exclusive events (c) Three non-mutually
exclusive events
The Special Addition Rule
If event A and event B are mutually exclusive, then
P A or B  P A  PB
More generally, if events A, B, C, … are mutually exclusive, then
P A or B or C ...  P A  PB  PC  ...
That is, for mutually exclusive events, the probability that at least one of
the events occurs is equal to the sum of the individual probabilities.
Non-mutually exclusive events
The General Addition Rule
If A and B are any two events, then
P(A or B) = P(A) + P(B) – P(A & B).
In words, for any two events, the
probability that one or the other occurs
equals the sum of the individual
probabilities less the probability that both
occur.
P(A or B): Spade or Face Card
P (spade) + P (face card) – P (spade & face card) = 1/4 + 3/13 – 3/52
= 22/52
The Special Multiplication Rule (for independent events)
If events A, B, C, . . . are independent, then
P(A & B & C & ) = P(A) P(B) P(C).
What is the probability of all of these events occurring:
1. Flip a coin and get a head
2. Draw a card and get an ace
3. Throw a die and get a 1
P(A & B & C ) = P(A) · P(B) · P(C) = 1/2 X 1/13 X 1/6
Conditional Probability: For non-independent events
The probability that event B occurs given that event A has
occurred is called a conditional probability. It is denoted
by the symbol P(B | A), which is read “the probability of
B given A.” We call A the given event.
Contingency Table for Joint Probabilities
Contingency table for age and rank of faculty members
(using frequencies)
The Conditional-Probability Rule
If A and B are any two events, then
P( A & B )
P( B | A) 
.
P( A)
In words, for any two events, the conditional
probability that one event occurs given that the other
event has occurred equals the joint probability of the
two events divided by the probability of the given
event.
The ConditionalProbability Rule
P( A & B )
P( B | A) 
.
P( A)
P( R3 | A4 ) =
= 36/253
= 0.142
P( A4 | R3 ) =
= 36/320
= 0.112
Joint probability
distribution (using
proportions)
P( A & B )
P( B | A) 
.
P( A)
P( R3 | A4 ) =
= 0.031/0.217
= 0.142
P( A4 | R3 ) =
= 0.031/.0275
= 0.112
Contingency table of marital status and sex
(using proportions)
Joint probability
distribution (using
proportions)
P( A & B )
P( B | A) 
.
P( A)
The General Multiplication Rule
If A and B are any two events, then
P(A & B) = P(A) P(B | A).
In words, for any two events, their joint probability
equals the probability that one of the events occurs times
the conditional probability of the other event given that
event.
Note: Either
1) The events are independent and then
P(A & B) = P(A) · P(B).
Or
2) The events are not independent and then a
contingency table must be used
Independent Events
Event B is said to be independent of event A if the
occurrence of event A does not affect the probability that
event B occurs. In symbols,
P(B | A) = P(B).
This means that knowing whether event A has occurred
provides no probabilistic information about the
occurrence of event B.
Class
Fr
So
Ju
Se
Male
40
50
50
40
| 180
Female
80
100
100
80
| 360
120
150
150
120
| 540
Probability and the Normal Distribution
• What is the probability of randomly
selecting an individual with an I.Q. between
95 and 115? Mean 100, S.D. 15.
• Find the z-score for 95 and 115 and
compute the area between
More Preview of Experimental Design
Using probability to evaluate a treatment effect. Values that are extremely
unlikely to be obtained from the original population are viewed as
evidence of a treatment effect.
A Preview of Sampling Distributions
• What is the probability of randomly
selecting a sample of three individuals, all
of whom have an I.Q. of 135 or more?
So while the odds chance selection of a single person this far
• Find the z-score of 135, compute the tail
above the mean is not all that unlikely, the odds of a sample this
region
it to the 3rd power.
far above
theand
meanraise
are astronomical
z = 2.19
P = 0.0143
X
0.01433 = 0.0000029
X
• This concept is critical to understanding
future concepts
Summary
For multiple events there are two rules:
“AND” (multiplication) and “OR” (addition)
There are just a few special considerations:
1. For the “And” rule, if the events are not
independent, you don’t multiply, you use a
table.
2. For the “Or” rule, if the events are not
mutually exclusive you have to subtract off
their double count