Chapter 4
Probability
McGraw-Hill/Irwin
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
Probability
4.1
4.2
4.3
4.4
Probability and Sample Spaces
Probability and Events
Some Elementary Probability Rules
Conditional Probability and
Independence
4.5 Bayes’ Theorem (Optional)
4.6 Counting Rules (Optional)
4-2
LO4-1: Define a
probability and a sample
space.
4.1 Probability and Sample Spaces





An experiment is any process of observation with
an uncertain outcome
The possible outcomes for an experiment are called
the experimental outcomes
Probability is a measure of the chance that an
experimental outcome will occur when an
experiment is carried out
The sample space of an experiment is the set of all
possible experimental outcomes
The experimental outcomes in the sample space are
called sample space outcomes
4-3
LO4-1
Probability
If E is an experimental outcome, then P(E)
denotes the probability that E will occur
and:
Conditions
0  P(E)  1 such that:
1.


2.
If E can never occur, then P(E) = 0
If E is certain to occur, then P(E) = 1
The probabilities of all the experimental
outcomes must sum to 1
4-4
LO4-1
Assigning Probabilities to Sample Space
Outcomes
Classical method
1.
◦
For equally likely outcomes
Relative frequency method
2.
◦
Using the long run relative frequency
Subjective method
3.
◦
Assessment based on experience, expertise or
intuition
4-5
LO4-2: List the
outcomes
in a sample space and
use the list to compute
probabilities.



4.2 Probability and Events
An event is a set of sample space outcomes
The probability of an event is the sum of
the probabilities of the sample space
outcomes
If all outcomes equally likely, the probability
of an event is just the ratio of the number of
outcomes that correspond to the event
divided by the total number of outcomes
4-6
LO4-3: Use elementary
profitability rules to
compute probabilities.
4.3 Some Elementary Probability Rules
1.
2.
3.
4.
5.
6.
Complement
Union
Intersection
Addition
Conditional probability
Multiplication
4-7
LO4-4: Compute
conditional probabilities
and assess
independence.

4.4 Conditional Probability and
Independence
The probability of an event A, given that the
event B has occurred, is called the
conditional probability of A given B
◦ Denoted as P(A|B)

Further, P(A|B) = P(A∩B) / P(B)
◦ P(B) ≠ 0
4-8
LO4-5: Use Bayes’
Theorem to update prior
probabilities to posterior
probabilities (Optional).



4.5 Bayes’ Theorem
S1, S2, …, Sk represents k mutually exclusive
possible states of nature, one of which must
be true
P(S1), P(S2), …, P(Sk) represents the prior
probabilities of the k possible states of nature
If E is a particular outcome of an experiment
designed to determine which is the true state
of nature, then the posterior (or revised)
probability of a state Si, given the
experimental outcome E, is calculated using
the formula on the next slide
4-9
LO4-5
Bayes’ Theorem
Continued
P(Si  E)
P(Si|E) =
P(E)
P(Si )P(E|S i )

P(E)
P(Si )P(E|S i )

P(S1 )P(E|S1 )+P(S 2 )P(E|S 2 )+ ...+P(S k )P(E|S k )
4-10
LO4-6: Use elementary
counting rules to
compute probabilities
(Optional).

4.6 Counting Rules (Optional)
A counting rule for multiple-step
experiments
(n1)(n2)…(nk)

A counting rule for combinations
N!/n!(N-n)!
4-11