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Introduction to Probability
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1. Probability Terminology and Notation
- Experimental Definitions
- Two Examples
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2. Finding Probability of an Event
- Classical Method to Determine Probability
- Doubles or Evens?
- Empirical Method
- The Law of Large Numbers
- Compliment Probability
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3. Mutually Exclusive, Joint Probability, Independent
- Mutually Exclusive Events and the Addition Rule
- Joint Probability Tables
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4. Venn Diagrams and Conditional Probability
- Venn Diagrams using Set Notation & Words
- Shading Venn Diagrams
- Venn Diagrams and Notation
- Tree Diagrams and Venn Diagrams
- Word Problem connected to Venn Diagrams
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Introduction to Probability
DavidK, Oct 30, 2020
Introducing Probability in a Statistics Course
Table of Contents
- Probability Terminology and Notation
- Experimental Definitions
- Two Examples
- Finding Probability of an Event
- Classical Method to Determine Probability
- Doubles or Evens?
- Empirical Method
- The Law of Large Numbers
- Compliment Probability
- Mutually Exclusive, Joint Probability, Independent
- Mutually Exclusive Events and the Addition Rule
- Joint Probability Tables
- Venn Diagrams and Conditional Probability
- Venn Diagrams using Set Notation & Words
- Shading Venn Diagrams
- Venn Diagrams and Notation
- Tree Diagrams and Venn Diagrams
- Word Problem connected to Venn Diagrams
Experimental Definitions
experiment
An experiment is a planned operation carried out under controlled conditions or in a way that can be repeated.
outcome (or simple event)
An outcome (or simple event) is the exact result of an experiment usually written down to form the statistical data.
Event:
a certain set of outcomes under considerations. Usually denoted with a capital letter,
like E.
Probability
The probability of event E occurring is written P(E).
P(E) = 1, if we are certain that the event will occur.
P(E) = 0, if event E cannot happen.
P(E)= 0.5 =50% if event E is equally likely to occur or not occur.
Sample Space
The collection of all possible outcomes of the experiment. Often denoted by the Greek capital letter omega, Ω. Our textbook uses capital S.
Classical Method to Determine Probability
A Classic Example
You have a fair 6 sided dice. What is the probability that the roll will be prime?
Let E be the event that the dice is rolled is prime. We are trying to find P(E).
Each outcome in the sample space is equally likely so we can take the number of outcomes in E and divide it by the number of outcomes in the entire sample space to find P(E).
P(E)=
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1/2
Since the number of outcomes in E is 3 and the number of outcomes in the sample space is 6.
Notation. | { } |
Mathematicians often use absolute value vertical bars to denote the number of elements in a set.
Find |{blue,red,green,yellow}|.
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4
Classical Method
The classic method for finding probabilities is only used when each outcome is equally likely.
To find the probability of an event E, take the number of outcomes in E and divide it by the total number of outcomes in the sample space.
Since an event can be considered the set of all outcomes it comprises we can write the classical method for finding the probability of event E as
Other Theoretical Methods
Later we will use Venn diagrams and the fact that P(Ω) = 1 to find probabilities.
We will also be using tree diagrams for finding probabilities when chance outcomes have intermediate stages or can be broken into discrete categories.
We will also learn the addition and product rule to find probabilities.
Sometimes the only way to find probability is by guess work and intuition, but you won't be asked to do that on a test.
Mutually Exclusive, Joint Probability, Independent
Create a table for joint probabilities and determine whether events are mutually exlusive, overlapping, independent, and dependent. Learn the probability properties these events have.
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1. Mutually Exclusive Events and the Addition Rule
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2. Joint Probability Tables
Mutually Exclusive Events and the Addition Rule
Definition Mutually Exclusive
Two events are mutually exclusive if they can never happen together.
If P(A and B)=0, then events A and B are mutually exclusive.
More Joe Cruelty
Let B stand for Joe getting hit by one or more buses, C for Joe getting hit by one or more cars, M for Joe getting hit by a motorcycle, T for Joe getting hit by one ore more different kinds of vehicle, S for Joe getting hit by several different kinds of vehicles, and L for Joe getting hit by no vehicles.
Are these mutually exclusive events?
You Create an Example
You throw a coin 5 times. Choose three events that are mutually exclusive and label them.
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Let A represent getting no heads. B represent getting two or three heads. C represents getting more than three heads.
Probability Addition to find P(A or B)
The addition rule for mutually exclusive events is easy. To find the probability that either A or B occurs add their individual probabilities together.Overlapping Events.
If two events are not mutually exclusive then they are overlapping events. For example, assume that at a high school with 300 you count the number of students who speak French and Spanish and make this Venn diagram. 
How many students speak French? What is the probability that a student speaks French?
Let F represent the event that a randomly chosen student speaks French.
How many students speak French? What is the probability that a student speaks French?
Let F represent the event that a randomly chosen student speaks French.Font sizeFont size
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9+7 = 16 = |F|
P(F) = 16/300 = 0.05333
Let S represent the event that a randomly chosen student speaks Spanish.
Find P(S)
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7+14 = 21 = |S|
P(S) = 21/300 = 0.07
Find the probability that a student speaks both Spanish and French.
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P(S and F) = 7/300 = 0.02333333333
Notice this is not = 0
Find P(F or S) = P(F and ) + P(F and S) + P( and S)
This comes easily from the Venn Diagram.
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P(F or S) = 9/300 + 7/300 + 14/300 = 30/300 = 0.1
Use the Addition Rule P(A or B) = P(A) + P(B) - P(A and B)
Find P(F or S) = P(F) + P(S) - P(F and S).
We subtract the trilingual speakers because otherwise we would be double counting them in this overlapping event.
This comes more easily from a "Klapheck" diagram.
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16/300 + 21/300 - 7/300 = 30/300 = 0.1
Why is the addition rule P(A or B) = P(A) + P(B) - P(A and B) still true for mutually exclusive events A and B?
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P(A or B) = P(A) + P(B) - P(A and B)
= P(A) + P(B)
since P(A and B) = 0 for mutually exclusive events.
Venn Diagrams using Set Notation & Words
Match the set notation descriptions with the shaded area shown in the Venn diagrams and the word descriptions.
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