Probability
Ray Block, Jr.
PS 585 (Research Methods)
Fall, 2003
Today’s Blueprint
Last Class: Statistics
-
What it “is”
-
What they “are”
-
How to Lie with them
-
How to tell the truth with them
Today’s Class: Probability
-
The Logic Behind It
-
[Some] Rules to It
The Logic Behind Probability
The Logic Behind It
-
What is probability?
-
A theory that began in 17th century France
-
Refers to the relative likelihood of outcomes occurring
Basic Definitions
-
Random Variables
-
Random Experiments
-
Elementary Outcomes
-
Sample Spaces
1) Random variable:
-
A variable whose outcome is uncertain until a random “experiment” is done
-
Example: A coin flip (the flip is the variable)
2) Random experiment
-
The process of observing a chance event
-
Example: Writing down a series of coin flips
3) Elementary outcomes:
-
All the possible values a random variable can take in a random experiment
-
Example: A flipped coin can either fall “heads” (H) or “tails” (T)
4) Sample Space:
-
The set of all elementary outcomes
-
Example: Flipping 2 coin gives you the following sample space: {H,T}.
-
Within that space are the following possible combinations {(H,H) (H,T)
(T,H) (T,T)}
Two approaches to thinking “probabilistically”:
-
Relative frequencies approach
-
Subjective approach
1) Relative frequencies approach:
-
Useful when you can that we can perform the “experiment” repeatedly under
similar conditions
-
Example: What is the likelihood of me making 5 out of 10 free throws?
2) Subjective approach:
-
Useful when you can that we can only perform the experiment once
-
Assign a value to the event that reflects your likelihood of the event
happening
-
Example: What is the likelihood of me getting an A in my class?
What is Probability (Take 2):
-
Probability is how we take a sample and test hypotheses about a population
-
Testing the probability that a random variable will take on some elementary
outcome
Some Rules of Probability
Some Probability Rules:
-
All probabilities are between 0 and 1 inclusive
-
The probability of an event which cannot occur is 0.
-
The probability of an event which must occur is 1.
-
The sum of all the probabilities in the sample space is 1
-
The probability of any event which is not in the sample space is zero.
Based on the above logic, we get the following rules:
-
The Converse Rule
-
The Addition Rule
-
The Multiplication Rule
The Converse Rule:
-
The probability of a specific elementary outcome not occurring is equal
to 1 minus the probability that it will occur
-
Example:
-
Pr(Coin Flip = Heads) = ½ = 0.5
-
Pr(Coin Flip = Tails) = ½ = 0.5
-
Therefore: Pr(Heads) = 1 – Pr(Tails) 1.0 - 0.5 = 0.5
The Addition Rule = The Probability of observing any one of several mutually
exclusive elementary outcomes is equal to the sum of their separate
probabilities
-
Mutually Exclusive = Independent Outcomes
-
The occurrence of one outcome does not influence the probability of another
outcome
-
Example: Pr(Card Draw = Ace of Spades OR Ace of Hearts) = 1/52 + 1/52 =
2/52 = 0.04
The Multiplication Rule
-
The probability of observing a combination of independent outcomes equals
the product of their separate probabilities
-
Example: Pr(Card Draw = Ace of Spades AND Ace of Hearts) = 1/52 * 1/52
= 1/2704 = .00037
References (FYI):
-
Levin, Jack and James Alan Fox. 2003. Elementary Statistics In Social Research,
9th Edition. Boston, MA: Pearson Education Group, Inc.
-
Wonnacott, Thomas H. and Ronald J. Wonnacott. 1990. Introductory Statistics,
5th Edition. New York, NY: John Wiley & Sons.
-
Stats: Introduction to Probability http://www.richland.cc.il.us/james/lecture/m170/ch05-int.html