The theory of probability

  Introduction
  Foundation
  Calculating
  Presize definition
  Binomial distribution
  Hypergeometric distribution
  Combined events
  Inseparable events
  'Ion Saliu's Paradox'
  Combinatorics
  Markov Chains
  Random Walks
  Contacts

We know a lot about probability if we can determine these two fundamental elements:
~ total possible cases, n;
~ number of favorable cases, N.

In general, the total number of possible cases is calculated by combinatorics, a branch of mathematics. Still at this web site, you'll find the best presentation of combinatorics. All four sets are clearly presented. Moreover, free software is provided to generate all possible types of sets and also calculate total possible elements in the sets — PermuteCombine.EXE. In other cases, calculating total number of cases is a matter of enumeration. How many sides does the coin have? Two = that's the number of total cases. A die has six faces = that's total possible cases.

Determining the number of favorable cases is more difficult in most cases. In a very simple case like betting on heads (not over head!) in coin tossing it is easy. There is one favorable case out of two. Betting on face six of a die is also easy: One favorable case (out of six).

Pascal's Triangle

In 1665, three years after his death, Pascal's famous arithmetical triangle was published. Pascal's Triangle is a well known mathematical pattern. This pattern was actually discovered in China, but it has been named after the first westerner to study it. According to Yunze He, "Pascal's" triangle was first developed during the Song Dynasty by a mathematician named Hue Yang.

The triangle is easily compiled.  Each line is formed by adding together each pair of adjacent numbers in the line above.  The first thing to notice about the triangle is how neatly line 5 summarises the five tosses of a coin (there are a total of 32 possible results of which one contains no heads, five contain 1 head, ten contain 2 heads, ten contain 3 heads, five contain 4 heads and one contains 5 heads). The Triangle is of great interest to gamblers, and provides the answer to questions relating to equipartition and combinations.