1.2  Probability Space

Definition 1.2.1. A probability space consists of a sample space \(\Omega\) and a probability function which takes an event as input and returns a real number between \(0\) and \(1\) as output satisfying the following axioms:

  1. the probability function satisfies \(P(\varnothing) = 0\) and \(P(\Omega) = 1\);

  2. if \(A_1,A_2,\ldots\) are mutually exclusive events, then

    \[P\lowparen{\bigcup_{i=1}^{\infty}A_i} = \sum_{i=1}^{\infty} P(A_i).\]

Example 1.2.2. When all outcomes in a finite sample space are equally likely, the probability of an event may be defined to be the number of outcomes it contains divided by the total number of outcomes. We now check whether this definition satisfies the probability space axioms. Let \(\Omega\) be finite. Then we define a function \(P\) by

\[P(A) = \frac{|A|}{|\Omega|}\]

for all \(A\subseteq \Omega\). Then \(P(\varnothing) = 0\) and \(P(\Omega) = 1\). Furthermore, let \(A_1,A_2,\ldots\subseteq \Omega\) be mutually exclusive. Since \(\Omega\) is finite, at most \(|\Omega|\) of them are nonempty, so there is \(n\) such that \(A_i=\varnothing\) for all \(i>n\). Then we have

\[\begin{aligned} P\lowparen{\bigcup_{i=1}^{\infty}A_i} &= \frac{|A_1\cup \cdots \cup A_n|}{|\Omega|} \\ &= \frac{|A_1| + \cdots + |A_n|}{|\Omega|} \\ &= \frac{|A_1|}{|\Omega|} + \cdots + \frac{|A_n|}{|\Omega|}= \sum_{i=1}^{\infty} P(A_i). \end{aligned}\]

That is, (PS1) and (PS2) hold; hence \((\Omega, P)\) is a probability space.

Proposition 1.2.3. If \((\Omega,P)\) is a probability space and \(A,B\subseteq \Omega\) then

  1. \(P(A^c) = 1 -P(A)\);
  2. if \(A\subseteq B\), then \(P(A)\le P(B)\);
  3. \(P(A\cup B) = P(A) + P(B) - P(A\cap B)\).