Transcript Probability - McGraw Hill Higher Education

Chapter 4
Probability
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Probability
4.1
4.2
4.3
4.4
4.5
4.6
The Concept of Probability
Sample Spaces and Events
Some Elementary Probability Rules
Conditional Probability and Independence
Bayes’ Theorem (Optional)
Counting Rules (Optional)
4-2
LO 1: Explain what a
probability is.



4.1 The Concept of
Probability
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
4-3
LO1
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
LO1

Assigning Probabilities to
Experimental Outcomes
Classical Method


Long-run relative frequency


For equally likely outcomes
In the long run
Subjective

Assessment based on experience, expertise or
intuition
4-5
LO 2: List the outcomes
in a sample space and
use the list to compute
probabilities.


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4.2 Sample Spaces and
Events
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
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
LO 3: Use elementary
profitability rules to
compute probabilities.
1.
2.
3.
4.
5.
6.
4.3 Some Elementary
Probability Rules
Complement
Union
Intersection
Addition
Conditional probability
Multiplication
4-7
LO3
Union and Intersection

The union of A and B are elementary events
that belong to either A or B or both


Written as A  B
The intersection of A and B are elementary
events that belong to both A and B

Written as A ∩ B
4-8
LO3
The Addition Rule

If A and B are mutually exclusive, then the
probability that A or B (the union of A and B)
will occur is
P(AB) = P(A) + P(B)

If A and B are not mutually exclusive:
P(AB) = P(A) + P(B) – P(A∩B)
where P(A∩B) is the joint probability of A and
B both occurring together
4-9
LO 4: Compute
conditional probabilities
and assess
independence.

The probability of an event A, given that the
event B has occurred, is called the
conditional probability of A given B


4.4 Conditional Probability
and Independence
Denoted as P(A|B)
Further, P(A|B) = P(A∩B) / P(B)

P(B) ≠ 0
4-10
LO4
The General Multiplication
Rule
There are two ways to calculate P(A∩B)
Given any two events A and B


1.
2.
P(A∩B) = P(A) P(B|A) and
P(A∩B) = P(B) P(A|B)
4-11
LO4
The Multiplication Rule

The joint probability that A and B (the
intersection of A and B) will occur is
P(A∩B) = P(A) • P(B|A) = P(B) • P(A|B)

If A and B are independent, then the
probability that A and B will occur is:
P(A∩B) = P(A) • P(B) = P(B) • P(A)
4-12
LO 5: Use Bayes’
Theorem to update prior
probabilities to posterior
probabilities (optional).

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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
4-13
LO5
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(Sk )P(E|S k )
4-14
LO 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)!
4-15