Chapter 4: Probability and Probability Models
Chapter 4: Probability and Probability Models
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
4.6: Counting Rules
Why Should You Care: Probabilities
Applications of probability in everyday life include:
Weather forecasts aiding in daily planning.
Choosing a study major based on anticipated job probabilities.
Guiding decisions in investments influenced by probabilities.
Assessing risks for diseases through test probability analyses.
Spam detection methods leveraging probability models.
4.1 Probability, Sample Spaces, and Probability Models
Definitions:
Experiment: A process of observation with uncertain outcomes.
Experimental Outcomes: The possible outcomes from an experiment.
Probability of an Event: A measure of the chance of an experimental outcome during an experiment.
Sample Space: The set of all potential outcomes of the experiment. Examples include
Tossing a coin: possible outcomes are heads and tails.
Sample Space Outcomes: The outcomes defined within the sample space.
What are Probabilities
Example:
A bag contains equal numbers of red and blue marbles.
Probability of drawing a red marble:
P(red) = 0.5Probability of drawing a blue marble:
P(blue) = 0.5
Concept of Long-Run Relative Frequency:
The interpretation of probability as expectation of long-term frequency.
Example with coin tossing: expected chance of heads is 50%, yet short runs may yield tails.
Long-run perspective clarifies the expected outcomes after a large number of trials.
Assigning Probabilities to Sample Space Outcomes
Event: A set of sample space outcomes.
Probability of an Event: Calculated as the sum of probabilities of relevant sample space outcomes.
Methods to Assign Probabilities:
Classical Method: Applicable for equally likely outcomes.
Relative Frequency Method: Utilizes outcomes from repeated experiments.
Subjective Method: Based on subjective assessment, experience, or expertise.
Conditions for Assigning Probabilities
Essential Conditions:
Probabilities assigned must satisfy: 0 \leq P(E) \leq 1
If event E cannot occur: P(E) = 0
If event E is certain to occur: P(E) = 1
The total probability of all experimental outcomes must add up to 1:
ext{Sum of all } P(E) = 1
1- Classical Method
Example: Tossing a fair coin has two equally likely outcomes:
Heads (H) and Tails (T).
Probability assignments:
P(H) = P(T) = \frac{1}{2} = 0.5
Rolling a Die:
Sample Space Outcomes: {1, 2, 3, 4, 5, 6}
Each outcome probability:
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \frac{1}{6}
2- Relative Frequency Method
Description: When classical assignment is challenging, estimate probability through experimentation over multiple trials.
Example:
Toss a coin 10 times, obtaining 6 heads.
Relative frequency of heads:
\text{Relative frequency} = \frac{6}{10} = 0.6
3- Subjective Method
Description: When experiments are impractical to repeat, probabilities estimated from intuition or expertise.
Example:
Selection of a new CEO among four candidates with subjective probabilities assigned:
P(A) = 0.1, P(C) = 0.2, P(H) = 0.5, P(R) = 0.2
Probability of selecting an internal candidate:
P(internal) = P(A) + P(H) = 0.1 + 0.5 = 0.6
Probability Models
Definition: A mathematical representation of random phenomena (experiments).
Key Components:
Random Variable: A numeric variable determined by an experiment's outcome.
Probability Distribution:
Specifies possible random variable values.
Provides table, graph, or formula for calculating probabilities.
Types of distributions:
Discrete: Binomial, Poisson.
Continuous: Normal, Exponential.
4.2 Probability and Events
Definition of an Event: A set of one or more outcomes from sample space.
Example: Tossing a die:
Event: “At least five spots will show.”
Corresponding sample space outcomes: {5, 6}.
Probability of the event:
P(event) = P(5) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
Exercise 4.1
Scenario: A couple aims to have two children.
Possible outcomes (sample space): {BB, BG, GB, GG}.
Assumed equal likelihood for each gender:
P(BB) = P(BG) = P(GB) = P(GG) = \frac{1}{4}
Probabilities for Specific Events:
Two boys: P(BB) = \frac{1}{4}
Two girls: P(GG) = \frac{1}{4}
One boy and one girl: P(BG) + P(GB) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}
At least one girl: P(BG) + P(GB) + P(GG) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}
First child is a boy: P(BG) + P(BB) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}
4.3 Some Elementary Probability Rules
Key Rules include:
Rule of Complements.
Intersection: Joint occurrence of two events.
Union: At least one of the events occurs.
Mutually Exclusive Events.
The Addition Rule.
Conditional Probability.
Multiplication Rule.
The Rule of Complements
Definition: The complement of an event A (denoted as Ā) comprises all outcomes not corresponding to A.
Mathematical representation:
Probability A not occurring: P(Ā) = 1 - P(A)
Application: For events, at least one of A or Ā occurs.
Exercise: Complement
Scenario: Rolling a fair die where the event of interest is obtaining a 6.
Complement Outcomes: Getting a 1, 2, 3, 4, or 5.
Probability Calculation:
P(Ā) = 1 - P(6) = 1 - \frac{1}{6} = \frac{5}{6}
The Rule of Intersection
Definition: The intersection of events A and B (A ∩ B) occurs when both A and B happen together.
Probability of Intersection:
P(A ∩ B)
Example 4.4: The Crystal Cable Case
Scenario Description: The company provides various household services; number of reachable households is 27.4 million, with 12.4 million being cable television customers.
Calculating Probabilities:
Probability of passing having television service:
P(A) = \frac{12.4}{27.4} = 0.45Probability of not having television service:
P(Ā) = 1 - P(A) = 1 - 0.45 = 0.55
Example Calculations From The Crystal Cable Case
Internet Service Probability:
Event B: Probability of having internet service:
P(B) = \frac{9.8}{27.4} = 0.36Probability of not having internet service calculated as:
Method 1:
P(B̄) = 1 - 0.36 = 0.64
Method 2:
P(B̄) = \frac{17.6}{27.4} = 0.64
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Probability of both internet and television service:
P(A ∩ B) = \frac{6.5}{27.4} = 0.24
Union of Events
Definition: Union A ∪ B occurs if either A or B (or both) happens.
Probability Calculation for Not Mutually Exclusive Events:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Example Union Calculation
For the Crystal Cable case:
At least one service:
P(A ∪ B) = 0.45 + 0.35 - 0.24 = 0.57
EventProbability with Neither Service
Probability of neither service:
Method 1:
P(Ā ∩ B̄) = \frac{11.7}{27.4} = 0.43Method 2:
P(Ā ∩ B̄) = 1 - P(A ∪ B) = 1 - 0.57 = 0.43
Mutually Exclusive Events
Definition: Events that cannot happen at the same time.
For example, if events A and B cannot both occur,
Probability of A or B (mutually exclusive):
P(A ∪ B) = P(A) + P(B)
Addition Rule for N Mutually Exclusive Events
Description: Allows summation over multiple mutually exclusive events.
Formula:
P(A1 ∪ A2 ∪ … ∪ AN) = P(A1) + P(A2) + … + P(AN)
4.4 Conditional Probability
Definition: The probability of an event A, given that event B has occurred, denoted as P(A|B).
Calculation:
Formula:
P(A|B) = \frac{P(A ∩ B)}{P(B)}
Example Conditional Probability Calculation
Scenario: Finding probability of having television service given internet service.
Calculation:
Method 1:
P(A|B) = \frac{6.5}{9.8} = 0.66
Method 2:
P(A|B) = \frac{P(A ∩ B)}{P(B)} = \frac{0.24}{0.36} = 0.66
General Multiplication Rule
Definition: Method to calculate joint probability.
Two ways to find joint probability:
P(A ∩ B) = P(A) imes P(B|A)
P(A ∩ B) = P(B) imes P(A|B)
Independent Events
Definition: Events A and B are independent if occurrence of one does not affect the other.
Conditions for Independence:
Either:
P(A|B) = P(A) or P(B|A) = P(B)
Multiplication Rule for Two Independent Events
When A and B are independent, the joint probability becomes:
P(A ∩ B) = P(A) imes P(B)
4.5 Bayes’ Theorem
Description: Updates prior probabilities of an event based on new evidence to determine posterior probabilities.
Theoretical Framework:
Prior probabilities: P(S1), P(S2), …, P(S_k)
Experimental outcome: E to determine state of nature.
Posterior calculation:
P(Si|E) = \frac{P(Si ∩ E)}{P(E)} = \frac{P(E|Si) imes P(Si)}{P(E)}
Example of Bayes’ Theorem
Scenario: Oil drilling decisions based on geological characteristics of a site.
States of Nature:
No oil: P(S_1) = 0.7
Some oil: P(S_2) = 0.2
Much oil: P(S_3) = 0.1
Reading Probability Reductions:
High reading intervals characterized by historical performance.
Final Calculation from Bayes’ Theorem
Given a high reading, posterior probabilities revised to approximately 0.21875 for no oil, 0.03125 for some oil, and 0.75 for much oil suggesting a drilling decision.
4.6 Counting Rules
General Formula for Sequential Outcomes:
If unique outputs exist at each step:
(n1)(n2)…(n_k)
Example in Quiz Context:
Three true/false questions yield 2 possibilities each.
Total outcomes:
(2)(2)(2) = 8
Counting Rules for Combination
Combination Calculation Formula: C(N, n) = \frac{N!}{n!(N − n)!}
Where $N! = N(N−1)(N−2)⋯1$ and $n! = n(n−1)(n−2)⋯1$ with special case $0! = 1$.
Example: Choosing 3 items from 6 stocks yields:
C(6, 3) = \frac{6!}{3!(6−3)!} = \frac{6×5×4}{3×2×1} = 20