1 Introduction to Probability

Wednesday, January 04, 2012

11:29 AM

1. A simple classification of real world phenomena

 

 Deterministic

 

Outcomes can be predicted precisely from an initial set of conditions.

"If you give me a set of conditions, I can provide you with the outcome."

 

Eg) finding the 3rd angle of a triangle.

 

 

 Random (Stochastic)

 

Outcomes cannot be predicted precisely from an initial set of conditions.

 

Eg) Rolling a dice

 

 

Many applications:

• Statistics
• (The science of data)
• Actuarial Science
• Calculating risk
• Computer Science
• OCR
• Engineering

 

Implication to daily life.

 

(Jan 05)

 

1.  Random events

Possible outcomes of a random trial/process/experiment

 

Eg

Randomly take a hand of 13 cards from a standard deck of 52 (13 Diamonds, 13 Hearts, 13 Spades, 13 Clubs)

 

A = "The hand consists of all 13 diamonds"

B = "The hand has at least 1 diamond"

 

Observations

• Run the trial only once, any event will either occur or not occur, but not both!
• Repeat the trial many times (under the same set of conditions).

B appears more often than A!

B is more likely to occur!

 

Definition 1

 

Probability (Prob.)

A quantitative measurement on how likely a random event will occur under repeated trials.

 

2.  The relative frequency of an event.

 



• 

 

• 

 

Let



• 
• 

 

Let



 



 

Eg.

 

Flip a coin.

 



 



 



 





 



 



 

0.5

 

Definition of prob. based on the stability of the relative frequency:

 

Definition 2

 

Probability

 



 

 

 

 

3.  Frequency interpretation of prob.

 



 

• 

"0.2" has no meaning.

 

• 

 

Eg.

 





 





 

Eg.

 



 



 

 

4.  Subjective prob.

Arbitrary measure of confidence.

 

Eg.

What is the probability that a terrorist crashed a plane on the twin towers on 9/11?

What is the probability that the moon landing happened?

 

5.  Classic definition of prob.

 

(A special case of Definition 2; Chapters 2 & 3)