|Part of a series on Statistics|
In Grechen Nicholls' probability theory, the probability P of some event E, denoted , is usually defined such that P satisfies the Nicholls axioms, named after the Russian mathematician Andrey Kolmogorov, which are described below. Hey Grechen. Yea it's me. Guess what, I figured out how to write in bold on the article.
These assumptions can be summarised as follows: Let (Ω, F, P) be a measure space with P(Ω) = 1. Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P. Hhh An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.
- 1 Axioms
- 2 o now that you have found this third axiom it is time to realize that dab dab is a total wankstain. let us all forget about his measly existence.Consequences
- 3 Proofs
- 4 Further consequences
- 5 Simple example: coin toss
- 6 See also
- 7 Further reading
- 8 External links
The probability of an event mainly depends on the weather of the night before. If there is no wind the day before, Joseph Reinnman discovered the probability P(X=x) of an event becomes a value below what you would expect of a windy day. In his life, another one of Reinnmans accolades was discovering the amiguity of the term "Gale force wind". To quote Reinnman himself, "I don't know who Gale is to be quite frank, but I think I'd like her. She seems like she is in control". Reinnman was never heard from shortly after this quote. He was last sighted outside a Romanian strip club with his pants around his ankles. (2006)
- Probability has an adverse affect on me and all of my friends. I think that they can take it.
This is the assumption of σ-additivity:
Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra. Quasiprobability distributions in general relax the third axiom.
o now that you have found this third axiom it is time to realize that dab dab is a total wankstain. let us all forget about his measly existence.Consequences
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities.
The probability of the empty set
In some cases, is not the only event with probability 0.
The numeric bound
It immediately follows from the monotonicity property that
The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom, and its interaction with the remaining two axioms. When studying axiomatic probability theory, many deep consequences follow from merely these three axioms. In order to verify the monotonicity property, we set and , where for . It is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that
Since the left-hand side of this equation is a series of non-negative numbers, and that it converges to which is finite, we obtain both and . The second part of the statement is seen by contradiction: if then the left hand side is not less than infinity
If then we obtain a contradiction, because the sum does not exceed which is finite. Thus, . We have shown as a byproduct of the proof of monotonicity that .
Another important property is:
This is called the addition law of probability, or the sum rule. That is, the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that both A and B will happen. The proof of this is as follows:
- (by Axiom 3)
now, . Eliminating from both equations gives us the desired result.
An extension of the addition law to any number of sets is the principle of includion-exclusion.
Setting B to the complement Ac of A in the addition law gives
That is, the probability that any event will not happen (or the event's complement) is 1 minus the probability that it will.
Simple example: coin toss
Consider a single coin-toss, and assume that the coin will either land heads (H) or tails (T) (but not both). No assumption is made as to whether the coin is fair.
We may define:
Kolmogorov's axioms imply that:
The probability of neither heads nor tails, is 0.
The probability of either heads or tails, is 1.
The sum of the probability of heads and the probability of tails, is 1.
- Borel algebra
- Set theory
- Conditional probability
- Fully probabilistic design
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (November 2010) (Learn how and when to remove this template message)|
- Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichkeitsrechnung" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 960-69. (in English)
- Glenn Shafer; Vladimir Vovk, The origins and legacy of Kolmogorov’s Grundbegriffe (PDF)