Axioms and Two Useful Theorems of Discrete Probability Functions

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The first article in this series, Introducing Probability Theory without Statistics, noted that probability distribution might be “discrete” or “continuous.”

This article builds the foundation for discrete probability functions, by introducing the four axioms and deriving two useful theorems from them.

Discrete Probability Functions: The Soul of Discretion

Image of a One in a Billion Probability by Micah Sittig

One in a Billion Probability: Image by Micah Sittig

The phrase, “probability distribution”, refers to the way the probability of an event is distributed over the outcomes. For example, the chance of a fair coin landing “heads” is 50%, as is the chance of “tails” is an example of a probability distribution. A different example would be the longevity of a specific model of light bulb under test conditions.

A “discrete” probability distribution has specific distinct outcomes. Here are two examples: there are 2 possible outcomes for a coin toss; there are 52 different playing cards (excluding jokers) in a standard deck.

However, there may be a countably infinite number of distinct outcomes for a discrete probability distribution. A simple example is the number of requests in a queue. Theoretically this could be an infinite number. (It feels infinite to the victim of an Internet “denial of service” attack, where someone maliciously generates requests for a Web page faster than the host can respond).

Axioms for Discrete Probability Functions

The “probability space”, ‘S’, is the set of all possible outcomes from an event. We can represent specific outcomes as the set ‘A’ or a group of sets Ai; all are subsets of ‘S’.

A specific probability function, ‘p(A)’, maps from a specific group of sets to the interval [0, 1] of the Real numbers. This branch of mathematics has four axioms.

  1. p({}) = p(Ø) = zero;
  2. P(S) = 1
  3. zero ≤ p(A) ≤ one for every outcome ‘A’ in ‘S’
  4. Let Ai be a group of two or more “disjoint sets” (there are no common elements in any two of these sets), then p(U(Ai)) = ∑ p(Ai) for all ‘i’

*The first axiom sets the probability of a null outcome, or an impossible outcome, to zero.
*The second axiom guarantees that some outcome found in the set ‘S’ will occur. So the probability of an outcome, as found in ‘S’, is one.
*The third axiom ensures that no outcome has a negative probability, and no outcome can be more certain than “certain”.
*The fourth axiom says that the probability of the union of two or more subsets Ai is the sum of the probabilities of each Ai.

Let’s give an example for the final, fourth axiom. Consider rolling one six-sided die, with outcomes S = {1, 2, 3, 4, 5, 6}. Each individual outcome has a 1/6 chance of appearing; so p({1}) = p({3}) = 1/6.

Since {1} and {3} are mutually exclusive, p({1, 3}) = p({1}) + p({3}) = 2/6 = 1/3.

On the other hand, p({1, 3}) and p({3, 5}) are not mutually exclusive, since ‘3’ is common. So p({1, 3, 5}) = 1/2, which is not equal to p({1, 3}) + p({3, 5}) = 2/3.

Click for Page Two: Discrete Probability Axiom Theorems

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