Probability Axioms and Theorems

Date: 04/10/2001 at 23:06:34
From: George Strickland
Subject: Laws of Probability

I've searched all over the net for the four laws of probability.  
I keep getting complicated math equations. Can you give me the four 
laws in simple-to-understand written words?

I definitely need the first law.

Thanks,
George Strickland

Date: 04/11/2001 at 13:40:05
From: Doctor Twe
Subject: Re: Laws of Probability

Hi George - thanks for writing to Dr. Math.

I'm not sure what four "laws" you're referring to. In math and 
science, a "law" can be an axiom, a postulate, or a theorem.

There are three axioms of probability - perhaps these are what you 
were thinking of:

Axiom 1: 
The probability of any event is greater than or equal to zero. 
(That is, there's no such thing as a "negative probability.")

   P(A) >= 0

Axiom 2: 
The probability of the sample space is 1. 
(That is, it is certain that one of the possible outcomes will occur.)

   P(S) = 1

Axiom 3: 
If A_1, A_2, A_3, ... is a sequence of mutually exclusive events 
(i.e. the joint occurrence of any pair of them is impossible), then:

   P(U A_i for i = 1 -> oo) = SUM (i = 1 -> oo) [ P(A_i) ]

This says that we can find the probability of one of several mutually 
exclusive events occurring by adding the probabilities of each event.

From these three axioms, we can derive many useful theorems, like:

Theorem: 
The probability of the empty set is zero. 
(That is, there is no chance of getting no outcome.)

Theorem: 
If sample space S has N equally likely outcomes, then the probability 
of event A is:

   P(A) = N(A) / N

where N(A) is the number of outcomes in A.

And so on. Perhaps if you give me more information of the "four laws" 
you refer to, I can be of more help.

If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/