Chapter 4

Probability

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Probability

4.1 Probability and Sample Spaces

4.2 Probability and Events

4.3 Some Elementary Probability Rules

4.4 Conditional Probability and Independence

4.5 Bayes’ Theorem (Optional)

4.6 Counting Rules (Optional)

4-2

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

LO4-1: Define a probability and a sample space.

4-3

Probability

If E is an experimental outcome, then P(E) denotes the probability that E will occur and:

Conditions1. 0 P(E) 1 such that:

If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1

2. The probabilities of all the experimental outcomes must sum to 1

LO4-1

4-4

Assigning Probabilities to Sample Space Outcomes

1. Classical method◦ For equally likely outcomes

2. Relative frequency method◦ Using the long run relative frequency

3. Subjective method◦ Assessment based on experience, expertise or

intuition

LO4-1

4-5

4.2 Probability and Events

An event is a set of sample space outcomesThe 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

LO4-2: List the outcomesin a sample space and use the list to compute probabilities.

4-6

4.3 Some Elementary Probability Rules

1. Complement

2. Union

3. Intersection

4. Addition

5. Conditional probability

6. Multiplication

LO4-3: Use elementary profitability rules to compute probabilities.

4-7

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

LO4-4: Compute conditional probabilities and assess independence.

4-8

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

LO4-5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (Optional).

4-9

Bayes’ Theorem Continued

))P(E|S+P(S...)+)P(E|S)+P(S)P(E|SP(S

))P(E|SP(S

P(E)

))P(E|SP(S

P(E)

E)P(S=|E)P(S

kk

ii

iiii

2211

LO4-5

4-10

4.6 Counting Rules (Optional)

A counting rule for multiple-step experiments

(n1)(n2)…(nk)

A counting rule for combinations

N!/n!(N-n)!

LO4-6: Use elementarycounting rules to compute probabilities (Optional).

4-11