Activity 1 - Advanced Statistical Concepts and Business Analytics
voyage
bowerman_9e_chap_041.pptxChapter 4
Probability and Probability Models
Copyright ©2018 McGraw-Hill Education. All rights reserved.
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Chapter Outline
4.1 Probability, Sample Spaces, and Probability Models
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)
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4.1 Probability, Sample Spaces, and Probability Models
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, a sample space, and a probability model.
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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:
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
LO4-1
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Assigning Probabilities to Sample Space Outcomes
Classical method
For equally likely outcomes
Relative frequency method
Using the long run relative frequency
Subjective method
Assessment based on experience, expertise or intuition
LO4-1
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LO4-1
Probability Models
Probability model: a mathematical representation of a random phenomenon
Random variable: a variable whose value is numeric and is determined by the outcome of an experiment
Probability distribution: A probability model describing a random variable
Discrete probability distributions (Chapter 6)
Continuous probability distributions (Chapter 7)
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LO4-1
Some Important Probability Distributions
Discrete probability distributions
Binomial distribution
Poisson distribution
Continuous probability distributions
Normal distribution
Exponential distribution
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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
LO4-2: List the outcomes in a sample space and use the list to compute probabilities.
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LO4-2
Classical Method
Example 4.1
A newly married couple plans to have two children
Would like to know all possible outcomes
BB BG GB GG
Want to know probabilities
Assuming all equal
P(BB) = P(BG) = P(GB) = P(GG) = ¼
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LO4-2
Subjective
Example 4.3
A company is choosing a new CEO
There are four candidates
Adams (A)
Chung (C)
Hill (H)
Rankin (R)
An industry analysts feels the probabilities are:
P(A) = 0.1
P(C) = 0.2
P(H) = 0.5
P(R) = 0.2
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4.3 Some Elementary Probability Rules
Complement
Union
Intersection
Addition
Conditional probability
Multiplication
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LO4-3: Use elementary profitability rules to compute probabilities.
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LO4-3
Complement
Figure 4.3
The complement () of an event A is the set of all sample space outcomes not in A
P() = 1 – P(A)
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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
LO4-3
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LO4-3
Some Elementary Probability Rules
Figure 4.4
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LO4-3
Mutually Exclusive
Figure 4.5
A and B are mutually exclusive if they have no sample space outcomes in common
In other words: P(A ∩ B) = 0
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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
LO4-3
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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.
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The General Multiplication Rule
There are two ways to calculate P(A ∩ B)
Given any two events A and B
P(A ∩ B) = P(A) P(B|A) and
P(A ∩ B) = P(B) P(A|B)
LO4-4
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Interpretation
Restrict sample space to just event B
The conditional probability P(A|B) is the chance of event A occurring in this new sample space
In other words, if B occurred, then what is the chance of A occurring
LO4-4
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Independence of Events
Two events A and B are said to be independent if and only if: P(A|B) = P(A)
This is equivalent to P(B|A) = P(B)
Assumes P(A) and P(B) greater than zero
LO4-4
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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)
LO4-4
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LO4-4
Contingency Tables
Table 4.3
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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
LO4-5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (Optional).
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Bayes’ Theorem Continued
LO4-5
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LO4-5
Example 4.14 The Oil Drilling Case: Site Selection
Example 4.14
Oil company trying to decide about drilling
There are three states of nature
No oil (S1) P(S1) = 0.7
Some oil (S2) P(S2) = 0.2
Much oil (S3) P(S3) = 0.1
Company can perform a seismic experiment
Gives three readings, low, medium, and high
P(high|none) = 0.04
P(high|some) = 0.02
P(high|much) = 0.96
Assume site gives high reading
Wish to revise the prior probabilities
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LO4-5
Example 4.14 The Oil Drilling Case: Site Selection Continued
Example 4.14
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4.6 Counting Rules (Optional)
A counting rule for multiple-step experiments (n1)(n2)…(nk)
A counting rule for combinations
LO4-6: Use some elementary counting rules to compute probabilities (Optional).
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LO4-6
A Tree Diagram of Answering Three True–False Questions
Figure 4.6
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