Chance operations & probability theory

From GenerativeArt

(Difference between revisions)
Jump to: navigation, search
(1st try at applet imbed)
(First version (eliminated applet experiment))
-
<nowiki>
+
== Probability of a Single Event ==
-
<div id="ballDistribution_container">
+
<SPAN STYLE="font-size: larger;">Introduction and examples </SPAN>
-
+
-
<!--[if !IE]> -->
+
-
<object classid="java:ballDistribution.class"
+
-
            type="application/x-java-applet"
+
-
            archive="http://www-viz.tamu.edu/students/ablev/genApps/ballDistribution.jar"
+
-
            width="400" height="700"
+
-
            standby="Loading Processing software..." >
+
-
           
+
-
<param name="archive" value="http://www-viz.tamu.edu/students/ablev/genApps/ballDistribution.jar" />
+
-
+
-
<param name="mayscript" value="true" />
+
-
<param name="scriptable" value="true" />
+
-
+
-
<param name="image" value="http://www-viz.tamu.edu/students/ablev/genApps/loading.gif" />
+
-
<param name="boxmessage" value="Loading Processing software..." />
+
-
<param name="boxbgcolor" value="#FFFFFF" />
+
-
+
-
<param name="test_string" value="outer" />
+
-
<!--<![endif]-->
+
-
+
-
<object classid="clsid:8AD9C840-044E-11D1-B3E9-00805F499D93"
+
-
codebase="http://java.sun.com/update/1.5.0/jinstall-1_5_0_15-windows-i586.cab"
+
-
width="400" height="700"
+
-
standby="Loading Processing software..."  >
+
-
+
-
<param name="code" value="ballDistribution" />
+
-
<param name="archive" value="http://www-viz.tamu.edu/students/ablev/genApps/ballDistribution.jar" />
+
-
+
-
<param name="mayscript" value="true" />
+
-
<param name="scriptable" value="true" />
+
-
+
-
<param name="image" value="http://www-viz.tamu.edu/students/ablev/genApps/loading.gif" />
+
-
<param name="boxmessage" value="Loading Processing software..." />
+
-
<param name="boxbgcolor" value="#FFFFFF" />
+
-
+
-
<param name="test_string" value="inner" />
+
-
+
-
<p>
+
-
<strong>
+
-
This browser does not have a Java Plug-in.
+
-
<br />
+
-
<a href="http://java.sun.com/products/plugin/downloads/index.html" title="Download Java Plug-in">
+
-
Get the latest Java Plug-in here.
+
-
</a>
+
-
</strong>
+
-
</p>
+
-
+
-
</object>
+
-
+
-
<!--[if !IE]> -->
+
-
</object>
+
-
<!--<![endif]-->
+
-
+
-
</div>
+
-
</nowiki>
+
 
 +
Given an ensemble of equally possible states, the probability of a given event occurring is the number of states which qualify as that event divided by the total number of possible states. A probability of 1 means the event is certain to occur. A probability of 0 means the event cannot possibly occur. A probability of .5 means the event is equally likely to occur or not occur.
 +
 
 +
 
 +
Example: What is the probability of flipping a coin and getting heads?
 +
 
 +
Total number of states in the ensemble = 2 (heads and tails)
 +
Total number of states that qualify as the event in question = 1 (heads)
 +
 
 +
Probability = 1/2 = .5
 +
 +
 
 +
Example: What is the probability of drawing an ace of spades from a shuffled deck of cards?
 +
 
 +
Total number of states in the ensemble = 52 (total number of cards in the deck)
 +
Total number of states that qualify as the event in question = 1 (the ace of spades)
 +
 
 +
Probability = 1/52 = .01923
 +
 +
 
 +
Example: What is the probability of drawing an ace from a shuffled deck of cards?
 +
 
 +
Total number of states in the ensemble = 52 (total number of cards in the deck)
 +
Total number of states that qualify as the event in question = 4 (the ace of spades, hearts, clubs, and diamonds)
 +
 
 +
Probability = 4/52 = .07692
 +
 
 +
 
 +
== Probability of a Sequence of Independent Events ==
 +
<SPAN STYLE="font-size: larger;">By example </SPAN>
 +
 
 +
 
 +
The probability of a sequence of independent probabilistic events is the product of the individual probabilities of those events.
 +
 
 +
 
 +
Example: What is the probability of flipping a coin and getting 4 heads in a row?
 +
 
 +
Probability of the individual event = .5
 +
Probability of a sequence of 4 of those events = .5 * .5 * .5 * .5 = .0625
 +
 
 +
Although it is counter-intuitive to some, this example and the previous one show that it's more likely you can draw an ace from a shuffled deck of cards (.07692) than flip 4 heads in a row (.0625).
 +
 
 +
 
 +
Example: What is the probability of drawing 4 aces from a shuffled deck of cards?
 +
 
 +
For the first card -
 +
Total number of states in the ensemble = 52 (total number of cards in the deck)
 +
Total number of states that qualify as the event in question = 4 (the ace of spades, hearts, clubs, and diamonds)
 +
 
 +
Probability = 4/52 = .07692
 +
 
 +
For the second card -
 +
Total number of states in the ensemble = 51 (total number of cards remaining in the deck)
 +
Total number of states that qualify as the event in question = 3 (the remaining aces)
 +
 
 +
Probability = 3/51 = .05882
 +
 
 +
For the third card -
 +
Total number of states in the ensemble = 50 (total number of cards remaining in the deck)
 +
Total number of states that qualify as the event in question = 4 (the remaining aces)
 +
 
 +
Probability = 2/50 = .04
 +
 
 +
For the fourth card -
 +
Total number of states in the ensemble = 49 (total number of cards remaining in the deck)
 +
Total number of states that qualify as the event in question = 4 (the remaining ace)
 +
 
 +
Probability = 1/49 = .02040
 +
 
 +
Probability of drawing 4 aces = (4/52) * (3/51) * (2/50) * (1/49) = .0000036919
 +
 
 +
 
 +
 
 +
== Combinatorial Probability ==
 +
<SPAN STYLE="font-size: larger;">Analyzing the qualifying states in an event</SPAN>
 +
 
 +
 
 +
As noted above the probability of an event is a function of the number of states in an ensemble and the number of those states which qualify as the event in question. In some situations counting the number of qualifying states is a matter of analyzing combinations.
 +
 
 +
 
 +
<SPAN STYLE="font-size: larger;">Pinball Example</SPAN>
 +
 
 +
In this example a ball is dropped on the top pin of a triangular arrangement of pins. The ball ends up exiting in one of several columns. The probability that a ball will end up in a given column is a function of the number of paths that lead to that column.
 +
 
 +
(Illustration and Applet pending)
 +
 
 +
 
 +
<SPAN STYLE="font-size: larger;">Pascal's Triangle</SPAN>
 +
 
 +
This geometric arrangement of numbers was devised by Blaise Pascal to demonstrate and calculate the probability of a proposition that is the combination of binary events.  To construct Pascal's Triangle each number is the sum of the 2 numbers in the line above.  The resulting numbers can be viewed as the expected statistical distribution.  For example, flipping 3 coins there is only one way to get all heads or all tails, but 3 ways to get 1 tail and 2 heads (THH, HTH, HHT), and 3 ways to get 1 head and 2 tails.
 +
 
 +
                            1
 +
                        1    1
 +
                      1    2  1
 +
                    1  3    3  1
 +
                  1  4    6    4  1
 +
                1  5  10  10  5 1
 +
              1  6  15 20  15  6 1
 +
            1  7  21 35 35 21  7  1
 +
          1  8 28 56 70 56 28  8  1
 +
 
 +
To calculate the probability of a given outcome one simply takes the number of combinations that can lead to the desired result, divided by the number of possible results.  So the probability of flipping 3 coins and  getting a head and 2 tails is:
 +
 
 +
(3) / ( 1 + 3 + 3 + 1 ) = 3/8 = .375
 +
 
 +
 
 +
The probability of flipping 6 coins and getting the most likely result of 3 heads and 3 tails is:
 +
 
 +
(20) / ( 1 + 6 + 15 + 20 + 15 + 6 + 1 ) = 20/64 = .3125
 +
 
 +
 
 +
Note that the most likely result, an equal number of heads and tails, happens less than than 1/3 of the time when flipping 6 coins!
 +
 
 +
 
 +
== The Independence of Events and the Gambler's Fallacy ==
 +
<SPAN STYLE="font-size: larger;">The myth of streaks </SPAN>
 +
 
 +
Its important to remember that each random event is unrelated to any other preceding event.  While flipping 10 heads in a row is very unlikely, once one has flipped 9 heads in a row the probability for the 10th flip is the same as it ever was, .5 for heads and .5 for tails.  Gamblers often think that so called "streaks" are meaningful, but in the case of random events they are not.  Oddly, for every misguided gambler who thinks that 9 heads in a row constitutes a streak that is likely to continue, there is usually another gambler who thinks 9 heads in a row means that tails are "due".  Both are wrong.
 +
 +
 
 +
== Statistical Distributions ==
 +
<SPAN STYLE="font-size: larger;">Different kinds of "random" </SPAN>
 +
 
 +
 
 +
(Text Pending)
 +
 
 +
 
 +
<SPAN STYLE="font-size: larger;">Uniform Distributions </SPAN>
 +
 
 +
 
 +
(Text and Illustration Pending)
 +
 
 +
 
 +
<SPAN STYLE="font-size: larger;">The Normal or Gaussian Distribution </SPAN>
 +
 
 +
 
 +
As more and more lines are added to Pascal's Triangle the distribution approximates a "bell curve" or the normal distribution, also known as a Gaussian Distribution.  This distribution is found throughout nature, for example height and weight measurements of a given population will approximate a normal distribution.
 +
 
 +
(Additional Text and Illustrations and Applet Pending)
 +
 
 +
 
 +
<SPAN STYLE="font-size: larger;">The Law of Large Numbers </SPAN>
 +
 
 +
 
 +
Even though events in nature will often tend towards a normal distribution, any particular set of measures or samples will show variation.  Where sample sizes are small the general distribution in nature may not be at all apparent.  As the sample size grows the apparent distribution will incrementally grow closer to the actual distribution in nature.
 +
 
 +
== Links ==
 +
 
 +
[[Category:Systems]]
 +
[[Category:Simple Highly Disordered Systems]]
 +
[[Category:Theory]]

Revision as of 08:33, 20 September 2008

Retrieved from "http://philipgalanter.com/generative_art/wiki/index.php5?title=Chance_operations_%26_probability_theory"
Personal tools