Difference between revisions of "Probability Review I"
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Let us look at two notions of the ''probability'' of an event happening: | Let us look at two notions of the ''probability'' of an event happening: | ||
| − | === The ''Frequentist'' | + | === 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>\left(x_1, x_2, \ldots, x_N\right)</math>, with the following assumptions: | ||
* No two events can occur simultaneously, and | * No two events can occur simultaneously, and | ||
| Line 14: | Line 14: | ||
{{NumBlk|::|<math>\sum_{i=1}^{N} P\left(x_i\right) = 1</math>|{{EquationRef|2}}}} | {{NumBlk|::|<math>\sum_{i=1}^{N} P\left(x_i\right) = 1</math>|{{EquationRef|2}}}} | ||
| − | === Observer Relative Probability === | + | === 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. | 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 == | == Probability Basics == | ||
| + | Recall that for two events, <math>A</math> and <math>B</math>: | ||
| + | |||
| + | {{NumBlk|::|<math>P\left(\overbar{A}\right) = 1 - P\left(A\right)</math>|{{EquationRef|3}}}} | ||
== Conditional Probability == | == Conditional Probability == | ||
Revision as of 14:28, 9 September 2020
Contents
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)
-
