Difference between revisions of "Probability Review I"

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=== The ''Frequentist'' Probability ===
 
=== The ''Frequentist'' Probability ===
Given <math>N</math> distinct possible events, <math>\left(x_1, x_2, \ldots, x_N\right)</math>, with the following assumptions:
+
Given <math>N</math> distinct possible events, <math>\{x_1, x_2, \ldots, x_N\}</math>, with the following assumptions:
 
* No two events can occur simultaneously, and  
 
* No two events can occur simultaneously, and  
* The events occur with frequencies <math>\left(n_1, n_2, \ldots, n_N\right)</math>,
+
* The events occur with frequencies <math>\{n_1, n_2, \ldots, n_N\}</math>,
 
Then the probability of an event <math>x_i</math> is given by:
 
Then the probability of an event <math>x_i</math> is given by:
  

Latest revision as of 17:18, 9 September 2020

Notions of Probability

Let us look at two notions of the probability of an event happening:

The Frequentist Probability

Given distinct possible events, , with the following assumptions:

  • No two events can occur simultaneously, and
  • The events occur with frequencies ,

Then the probability of an event is given by:

 

 

 

 

(1)

From this definition, we can see that:

 

 

 

 

(2)

The Observer Relative Probability

Probability is an assertion about the belief that a specific observer has of the occurrence of a specific event. Thus, here, two different observers may assign different probabilities to the same event or phenomenon. Additionally, the probability assigned to an event is likely to change as the observer learns more about an event, or if certain aspects of the event changes.

We can think of the observer relative probability of an event to be an approximation to the frequentist probability, and additionally, we can view new knowledge as stepping stones towards better estimates of the relative frequencies of an event.

Probability Basics

Recall that for two events, and :

 

 

 

 

(3)

 

 

 

 

(4)

Note that if , then and are mutually exclusive.

Conditional Probability

We define as the probability of occurring given that we know occurred. Thus, the joint probability of and is given by:

 

 

 

 

(5)

Since , we arrive at Bayes' Theorem:

 

 

 

 

(6)

Or equivalently:

 

 

 

 

(7)

If , then and are independent. From Bayes' Theorem, we get . Thus, for independent events,

 

 

 

 

(8)