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

If we define precisely the probabilistic events we can come up with a template of resolving just about every probability problem. I believe that the following two pairs of attributes are the essential foundation of the probability template:
~ separable – inseparable
~ single trial – multiple trials.

Separable – inseparable events
Is the coin separable in two sides? NOT! Therefore coin flipping is an inseparable event. Is the die separable in six faces? NOT! Therefore dice rolling is an inseparable event.
How about the lottery? Well, there are a number of balls inside a drawing machine. A number of balls (e.g. six) are separated from the total number of balls (e.g. 49). Therefore, a lottery drawing is a separable probability event.

Based on this attribute, we can calculate the probability differently. In general, in the first case we apply the very first formula of probability (FFPr = n/N) or the probability of the normal distribution (a more general and encompassing case than the first formula). In the separable case of probabilistic events, we calculate the probability by applying the hypergeometric distribution.

Single trial – multiple trials
If we toss the coin once = that's a single trial probability event. We roll the die once — also a single trial. The probability is calculated more easily in such situations. In general, the n/N formula is applied. One in two (1/2 or 0.5) is the probability to get heads in one coin toss.

The questions can get more and more complicated. That's how the so-called probability problems come to life. What is the probability to get heads five times in a row? What is the probability to roll face six of a die exactly 4 times in 10 throws? How about the probability to get at most 4 heads in 15 coin tosses? Or, what is the probability of getting at least 4 winners out of 6 numbers drawn in a 6/49 lotto game? To answer such questions we need to apply more sophisticated and complicated formulas and algorithms and even computer programs.

Converse - "Something that has been reversed; an opposite."

Often when you work out the probability of an event, you sometimes do not need  to work out the probability of an event occurring you need the opposite. The probability that the event will not occur. For example, The probability of throwing a 1 on a die is 1/6 therefore the probability of a 'non-1' is (1-1/6) which equals 5/6.

Converse probabilities are used to work out such problems such as, "What is the probability of exactly one soccer match ending in a draw within a group of  three separate matches?"

Let us assume the chance of a draw occurring in any match is 1/3 or 33.33%. To fulfill our target of only one match ending in a draw we would require the other matches to not end in a draw or (1-(1/3)) which equals 2/3 or 66.66%.

Therefore the probability of only one match out of three being drawn is 1/3x2/3x2/3 which equals = 4/27 or (.33*.67*.67) = 14.81%

In our group of three matches there are three ways for only one match to draw, DXX, XDX, XXD, therefore we need to add together all the probabilities, three in this case.

The final answer to the probability of one match drawing is (4/27)+(4/27)+(4/27) = 4/9 or (=.1481+.1481+.1481) = 44.44%.