1. Chapter 4
Probability

2. Chapter Outline
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)

3. 4.1 Probability and Sample Spaces
LO 4-1: 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 sample space
Also known as experimental outcomes and sample space outcomes
Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out

4. LO4-1
Probability
If E is a sample space outcome, then P(E) denotes the probability that E will occur and:
Conditions:
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
The probabilities of all the sample space outcomes must sum to 1

5. 4.2 Sample Spaces and Events
LO 4-2: List the outcomes in a sample space and use the list to compute probabilities.
4.2 Sample Spaces and Events
Sample Space: The set of all possible experimental outcomes
Sample Space Outcomes: The experimental outcomes in the sample space
Event: A set of sample space outcomes
Probability: The probability of an event is the sum of the probabilities of the sample space outcomes that correspond to the event

6. 4.3 Some Elementary Probability Rules
LO 4-3: Use elementary probability rules to compute probabilities.
4.3 Some Elementary Probability Rules
Complement
Union
Intersection
Addition
Conditional probability

7. Union and Intersection Diagram
LO4-3
Union and Intersection Diagram
Figure 4.4

8. LO4-3
The Addition Rule
If A and B are mutually exclusive, then the probability that A or 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

9. 4.4 Conditional Probability and Independence
LO 4-4: 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
Likewise, P(B|A) = P(A∩B) / P(A)

10. General Multiplication Rule
LO4-4
General Multiplication Rule
Given any two events, A and B
Referred to as general multiplication rule

11. 4.5 Bayes’ Theorem (Optional)
LO 4-5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (Optional).
4.5 Bayes’ Theorem (Optional)
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

12. Bayes’ Theorem Continued
LO4-5
Bayes’ Theorem Continued

13. 4.6 Counting Rules (Optional)
LO 4-6: 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)!