Introduction to Probability
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,[br]like E.
Probability
The probability of event E occurring is written P(E). [br]P(E) = 1, if we are certain that the event will occur. [br]P(E) = 0, if event E cannot happen.[br]P(E)= 0.5 =50% if event E is equally likely to occur or not occur.
Sample Space
[color=#ff0000][u]The collection of all possible outcomes of the experiment.[/u][/color] 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?[br]Let E be the event that the dice is rolled is prime. We are trying to find P(E).[br]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)=
Notation. | { } |
Mathematicians often use absolute value vertical bars to denote the number of elements in a set.
Find |{blue,red,green,yellow}|.
Classical Method
The classic method for finding probabilities is only used when each outcome is equally likely.[br]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.[br]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[br][math]P\left(E\right)=\frac{\left|E\right|}{\left|\Omega\right|}[/math]
Other Theoretical Methods
Later we will use Venn diagrams and the fact that P(Ω) = 1 to find probabilities.[br]We will also be using tree diagrams for finding probabilities when chance outcomes have intermediate stages or can be broken into discrete categories.[br]We will also learn the addition and product rule to find probabilities.[br]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 Events and the Addition Rule
Definition Mutually Exclusive
Two events are mutually exclusive if they can never happen together.[br]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.[br]Are these mutually exclusive events?[br]
You Create an Example
You throw a coin 5 times. Choose three events that are mutually exclusive and label them.
Probability Addition to find P(A or B)
[img]https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A9e38ae102a37cb33e00387372e33deeea9a01e74cca6ce89b19de92e%2BCOVER_PAGE_TINY%2BCOVER_PAGE_TINY.1[/img][br]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. [img]https://www.mathsisfun.com/data/images/set-language-ex2.svg[/img]How many students speak French? What is the probability that a student speaks French?[br]Let F represent the event that a randomly chosen student speaks French.
Let S represent the event that a randomly chosen student speaks Spanish.[br]Find P(S)
Find the probability that a student speaks both Spanish and French.
Find P(F or S) = P(F and [math]S^C[/math]) + P(F and S) + P([math]F^C[/math] and S)[br]This comes easily from the Venn Diagram.
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).[br]We subtract the trilingual speakers because otherwise we would be double counting them in this overlapping event.[br]This comes more easily from a "Klapheck" diagram.
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?
Venn Diagrams using Set Notation & Words
Match the set notation descriptions with the shaded area shown in the Venn diagrams and the word descriptions.