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Chapter 5
Section 1
Probability
Rules
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 1 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 2 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 3 of 33
Chapter 5 – Section 1
● Probability relates short-term results to long-term
results
● An example
 A short term result – what is the chance of getting a
proportion of 2/3 heads when flipping a coin 3 times
 A long term result – what is the long-term proportion
of heads after a great many flips
 A “fair” coin would yield heads 1/2 of the time – we
would like to use this theory in modeling
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 4 of 33
Chapter 5 – Section 1
● Relation between long-term and theory
 The long term proportion of heads after a great many
flips is 1/2
 This is called the Law of Large Numbers
● Relation between short-term and theory
 We can compute probabilities such as the chance of
getting a proportion of 2/3 heads when flipping a coin
3 times by using the theory
 This is the probability that we will study
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 5 of 33
Chapter 5 – Section 1
● Some definitions
 An experiment is a repeatable process where the
results are uncertain
 An outcome is one specific possible result
 The set of all possible outcomes is the sample space
● Example
 Experiment … roll a fair 6 sided die
 One of the outcomes … roll a “4”
 The sample space … roll a “1” or “2” or “3” or “4” or
“5” or “6”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 6 of 33
Chapter 5 – Section 1
● More definitions
 An event is a collection of possible outcomes … we
will use capital letters such as E for events
 Outcomes are also sometimes called simple events
… we will use lower case letters such as e for
outcomes / simple events
● Example (continued)
 One of the events … E = {roll an even number}
 E consists of the outcomes e2 = “roll a 2”, e4 = “roll a
4”, and e6 = “roll a 6” … we’ll write that as {2, 4, 6}
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 7 of 33
Chapter 5 – Section 1
● Summary of the example
 The experiment is rolling a die
 There are 6 possible outcomes, e1 = “rolling a 1”
which we’ll write as just {1}, e2 = “rolling a 2” or {2}, …
 The sample space is the collection of those 6
outcomes {1, 2, 3, 4, 5, 6}
 One event is E = “rolling an even number” is {2, 4, 6}
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 8 of 33
Chapter 5 – Section 1
● If E is an event, then we write P(E) as the
probability of the event E happening
● These probabilities must obey certain
mathematical rules
● We will be studying varying classes of
probabilities … these rules are true for all of
them
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 9 of 33
Chapter 5 – Section 1
● Rule – the probability of any event must be
greater than or equal to 0 and less than or equal
to 1
 It does not make sense to say that there is a –30%
chance of rain
 It does not make sense to say that there is a 140%
chance of rain
● Note – probabilities can be written as decimals
(0, 0.3, 1.0), or as percents (0%, 30%, 100%), or
as fractions (3/10)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 10 of 33
Chapter 5 – Section 1
● Rule – the sum of the probabilities of all the
outcomes must equal 1
 If we examine all possible cases, one of them must
happen
 It does not make sense to say that there are two
possibilities, one occurring with probability 20% and
the other with probability 50% (where did the other
30% go?)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 11 of 33
Chapter 5 – Section 1
● Probability models must satisfy both of these
rules
● There are some special types of events
 If an event is impossible, then its probability must be
equal to 0 (i.e. it can never happen)
 If an event is a certainty, then its probability must be
equal to 1 (i.e. it always happens)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 12 of 33
Chapter 5 – Section 1
● A more sophisticated concept
 An unusual event is one that has a low probability of
occurring
 This is not precise … how low is “low?
● Typically, probabilities of 5% or less are
considered low … events with probabilities of
5% or lower are considered unusual
● However, this cutoff point can vary by the
context of the problem
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 13 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 14 of 33
Chapter 5 – Section 1
● If we do not know the probability of a certain
event E, we can conduct a series of experiments
to approximate it by
frequ
of
E
P
(
E
)

numbe
of
trials
of
the
exp
● This becomes a good approximation for P(E) if
we have a large number of trials (the law of large
numbers)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 15 of 33
Chapter 5 – Section 1
● Example
● We wish to determine what proportion of
students at a certain school have type A blood
 We perform an experiment (a simple random sample!)
with 100 students
 If 29 of those students have type A blood, then we
would estimate that the proportion of students at this
school with type A blood is 29%
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 16 of 33
Chapter 5 – Section 1
● Example (continued)
● We wish to determine what proportion of
students at a certain school have type AB blood
 We perform an experiment (a simple random sample!)
with 100 students
 If 3 of those students have type AB blood, then we
would estimate that the proportion of students at this
school with type AB blood is 3%
 This would be an unusual event
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 17 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 18 of 33
Chapter 5 – Section 1
● The classical method applies to situations where
all possible outcomes have the same probability
● This is also called equally likely outcomes
● Examples
 Flipping a fair coin … two outcomes (heads and tails)
… both equally likely
 Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6)
… all equally likely
 Choosing one student out of 250 in a simple random
sample … 250 outcomes … all equally likely
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 19 of 33
Chapter 5 – Section 1
● Because all the outcomes are equally likely, then
each outcome occurs with probability 1/n where
n is the number of outcomes
● Examples
 Flipping a fair coin … two outcomes (heads and tails)
… each occurs with probability 1/2
 Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6)
… each occurs with probability 1/6
 Choosing one student out of 250 in a simple random
sample … 250 outcomes … each occurs with
probability 1/250
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 20 of 33
Chapter 5 – Section 1
● The general formula is
Numbe
of
ways
E
can
occ
P
(
E
)

Numbe
of
possi
outc
● If we have an experiment where
 There are n equally likely outcomes (i.e. N(S) = n)
 The event E consists of m of them (i.e. N(E) = m)
then
mN
(
E
)
P
(
E
)
 
n N
(
S
)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 21 of 33
Chapter 5 – Section 1
● Because we need to compute the “m” or the
N(E), classical methods are essentially methods
of counting
● These methods can be very complex!
● An easy example first
● For a die, the probability of rolling an even
number
 N(S) = 6 (6 total outcomes in the sample space)
 N(E) = 3 (3 outcomes for the event)
 P(E) = 3/6 or 1/2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 22 of 33
Chapter 5 – Section 1
● A more complex example
● Three students (Katherine, Michael, and Dana)
want to go to a concert but there are only two
tickets available
● Two of the three students are selected at
random
 What is the sample space of who goes?
 What is the probability that Katherine goes?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 23 of 33
Chapter 5 – Section 1
● Example continued
● We can draw a tree diagram to solve this
problem
● Who gets the first ticket? Any one of the three …
Katherine
Start
Michael
Dana
First ticket
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 24 of 33
Chapter 5 – Section 1
● Who gets the second ticket?
 If Katherine got the first, then either Michael or Dana
could get the second
Michael
Katherine
Dana
Start
Michael
Second ticket
Dana
First ticket
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 25 of 33
Chapter 5 – Section 1
● That leads to two possible outcomes
Outcomes
Michael
Katherine
Michael
Katherine
Dana
Start
Katherine
Dana
Michael
Second ticket
Dana
First ticket
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 26 of 33
Chapter 5 – Section 1
● We can fill out the rest of the tree
Michael
Katherine
Michael
Dana
Katherine
Dana
Katherine
Michael
Katherine
Dana
Michael
Dana
Katherine
Dana
Katherine
Michael
Dana
Michael
Katherine
Start
Michael
Dana
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 27 of 33
Chapter 5 – Section 1
● Katherine goes in 4 out of the 6 outcomes … a
4/6 or 2/3 probability
Katherine
Michael
Michael
Dana
Katherine
Dana
Katherine
Michael
Katherine
Dana
Michael
Dana
Katherine
Dana
Katherine
Michael
Dana
Michael
Katherine
Start
Michael
Dana
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 28 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 29 of 33
Chapter 5 – Section 1
● Sometimes probabilities are difficult to calculate,
but the experiment can be simulated on a
computer
● If we simulate the experiment multiple times,
then this is similar to the situation for the
empirical method
● We can use
freque
of
E
P
(
E
)

numbe
of
runs
of
the
sim
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 30 of 33
Chapter 5 – Section 1
● Learning objectives
1 Understand the rules of probabilities
2 Compute and interpret probabilities using the
empirical method
3 Compute and interpret probabilities using the
classical method
4 Use simulation to obtain data based on probabilities
5 Understand subjective probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 31 of 33
Chapter 5 – Section 1
● A subjective probability is a person’s estimate of
the chance of an event occurring
● This is based on personal judgment
● Subjective probabilities should be between 0
and 1, but may not obey all the laws of
probability
● For example, 90% of the people consider
themselves better than average drivers …
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 32 of 33
Summary: Chapter 5 – Section 1
● Probabilities describe the chances of events
occurring … events consisting of outcomes in a
sample space
● Probabilities must obey certain rules such as
always being greater than or equal to 0
● There are various ways to compute probabilities,
including empirically, using classical methods,
and by simulations
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 33 of 33
Examples
● Identify as Classical, Empirical, or Subjective
Probability:
 In his fall 1998 article in Chance Magazine, (“A
Statistician Reads the Sports Pages,” pp. 17-21,)
Hal Stern investigated the odds that a particular
horse will win a race. He reports that these odds
are based on the amount of money bet on each
horse. The odds can be used to calculate
probabilities. When a probability is given that a
particular horse will win a race, is this empirical,
classical, or subjective probability?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 34 of 33
Examples
● Identify as Classical, Empirical, or Subjective Probability:

Pass the Pigs TM is a Milton-Bradley game in which pigs are
used as dice. Points are earned based on the way the pigs
land. There are six possible outcomes when one pig is
tossed. A class of 52 students rolled pigs 3,939 times. The
number of times each outcome occurred is recorded in the
table at right. (Source:
http://www.members.tripod.com/~passpigs/prob.html) Are
these probabilities empirical, classical, or subjective?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 35 of 33
Examples
● Identify as Classical, Empirical, or Subjective
Probability:

In a draft lottery, balls representing each birthday
are placed in a bin and mixed. Individuals whose
birth date is drawn are selected for military
service. Ignore leap year. The probability that a
particular day, i.e. July 1, will be selected on the
first draw is 1/365. Is this an example of an
empirical, classical, or subjective probability?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 36 of 33
● Suppose two students are selected at random.
● What is the probability that the first student was
born in April?
● What is the probability that both students were
born in July?
● Is it likely that these two students share the same
birthday?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 37 of 33
● 30/365 ≈ 0.0822
● 961/133225 ≈ 0.0072
●
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 1 – Slide 38 of 33