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

 Probability starts with logic. There is a set of N elements. We can define a sub-set of n favorable elements, where n is less than or equal to N. Probability is defined as the rapport of the favorable cases over total cases, or calculated as:

It is the Fundamental Formula of Probability (FFPr) and everything in theory of probability is derived from it.

A simple example of sets and favorable elements: there are 10 balls in a jar; 5 of the 10 balls are red; what is the probability to extract one red ball from the jar? There 5 favorable cases in a total of 10 elements; the probability is 5/10 = Ѕ = 0.5.
The easiest case is coin tossing. What is the probability of heads in one coin toss: 1 / 2 = 0.5.
The probability can be also understood as expected number of successes in one trial. That formulation makes it easier to understand why probability can never be higher than 1: no event can have more than one success in one try! The flip side is that the phenomena with only one side do not exist. The Everything does have at least two sides. In fact, Everything is a unity of two opposites — the all encompassing Flipping Coin paradigm. That formulation makes it easier to understand why probability can never be higher than 1: no event can have more than one success in one try!

The term odds is often used in theory of probability, especially the branch dealing with gambling. The most correct usage of odds points to the degree of difficulty for an event to occur. The odds are N to n or (N – n) to n. It is widely used in horse racing. ”The odds for Horse X to win are 5 to 2”. The N to n case represents the odds against.

Another form of odds is used now even more widely than the term probability or probabilities. It is the so-called favorable odds. The favorable odds are calculated and expressed as n in N, most frequently as 1 in N1. If probability is most ofen expressed as a floating point number between 0 and 1, the favorable odds tend to be expressed as 1 in Number_Greater_Than_1, usually an integer. “The odds of winning 'exactly 3 of 6' in a 6/49 lotto game are '1 in 57'.”

Personally, I prefer expressing the probability as two terms: n/N and 1 in N1. I never use the expression N to n.

The law of large numbers / "The law of averages"

The theory of probability becomes of enhanced value to gamblers when it is used with the law of large numbers.  The law of large numbers states that:

“If the probability of a given outcome to an event is P and the event is repeated N times, then the larger N becomes, so the likelihood increases that the closer, in proportion, will be the occurrence of the given outcome to N*P.”

For example:-

If the probability of throwing a double-6 with two dice is 1/36, then the more times we throw the dice, the closer, in proportion, will be the number of double-6s thrown to of the total number of throws. This is, of course, what in everyday language is known as the law of averages.  The overlooking of the vital words 'in proportion' in the above definition leads to much misunderstanding among gamblers.  The 'gambler's fallacy' lies in the idea that “In the long run” chances will even out. Thus if a coin has been spun 100 times, and has landed 60 times head uppermost and 40 times tails, many gamblers will state that tails are now due for a run to get even.  There are fancy names for this belief.  The theory is called the maturity of chances, and the expected run of tails is known as a 'corrective', which will bring the total of tails eventually equal to the total of heads.  The belief is that the 'law' of averages really is a law which states that in the longest of long runs the totals of both heads and tails will eventually become equal.

In fact, the opposite is really the case.  As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

Let us return to our example of 60 heads and 40 tails in 100 spins, and imagine that the next 100 spins result in 56 heads and 44 tails.  The 'corrective' has set in, as the percentage of heads has now dropped from 60 per cent to 58 per cent.  But there are now 32 more heads than tails, where there were only 20 before.  The 'law of averages' follower who backed tails is 12 more tosses to the bad.  If the third hundred tosses result in 50 heads and 50 tails, the 'corrective' is still proceeding, as there are now 166 heads in 300 tosses, down to 55-33 per cent, but the tails backer is still 32 tosses behind.

Put another way, we would not be too surprised if after 100 tosses there were 60 per cent heads.  We would be astonished if after a million tosses there were still 60 per cent heads, as we would expect the deviation from 50 per cent to be much smaller.  Similarly, after 100 tosses, we are not too surprised that the difference between heads and tails is 20.  After a million tosses we would be very surprised to find that the difference was not very much larger than 20.

A chance event is uninfluenced by the events which have gone before.  If a true die has not shown 6 for 30 throws, the probability of a 6 is still 1/6 on the 31st throw.  One wonders if this simple idea offends some human instinct, because it is not difficult to find gambling experts who will agree with all the above remarks, and will express them themselves in books and articles, only to advocate elsewhere the principle of 'stepping in when a corrective is due'.

An understanding of the law of the large numbers leads to a realisation that what appear to be fantastic improbabilities are not remarkable at all but, merely to be expected.