1.1 Probability systems  (Page 3/4)

Definition

A probability system consists of a basic set Ω of elementary outcomes of a trial or experiment, a class of events as subsets of the basic space, and a probability measure $P(\cdot )$ which assigns values to the events in accordance with the following rules:

• (P1) For any event A , the probability $P\left(A\right)\ge 0$ .
• (P2) The probability of the sure event $P\left(\Omega \right)=1$ .
• (P3) Countable additivity . If $\{A_i:1\in J\}$ is a mutually exclusive, countable class of events, then the probability of the disjoint union is the sum of theindividual probabilities.

The necessity of the mutual exclusiveness (disjointedness) is illustrated in [link] . If the sets were not disjoint, probability would be counted more than once in the sum.A probability, as defined, is abstract—simply a number assigned to each set representing an event. But we can give it aninterpretation which helps to visualize the various patterns and relationships encountered. We may think of probability as mass assigned to an event. The total unit mass is assigned to the basic set Ω . The additivity property for disjoint sets makes the mass interpretation consistent. We can use this interpretation as a precise representation.Repeatedly we refer to the probability mass assigned a given set. The mass is proportional to the weight, so sometimes we speak informally of the weight rather than the mass.Now a mass assignment with three properties does not seem a very promising beginning. But we soon expand this rudimentary list of properties. We use the mass interpretation to helpvisualize the properties, but are primarily concerned to interpret them in terms of likelihoods.

• (P4) $P\left(A^c\right)=1-P\left(A\right)$ . This follows from additivity and the fact that
  $$1=P\left(\Omega \right)=P(A\bigvee A^c)=P\left(A\right)+P\left(A^c\right)$$
• (P5) $P\left(\varnothing \right)=0$ . The empty set represents an impossible event. It has no members, hence cannot occur. It seems reasonable that it should be assigned zero probability(mass). Since $\varnothing ={\Omega }^c$ , this follows logically from (P4) and (P2) .

  

• (P6) If $A\subset B$ , then $P\left(A\right)\le P\left(B\right)$ . From the mass point of view, every point in A is also in B , so that B must have at least as much mass as A . Now the relationship $A\subset B$ means that if A occurs, B must also. Hence B is at least as likely to occur as A . From a purely formal point of view, we have
  $$B=A\bigvee A^{c}B\text{so}\text{that}P\left(B\right)=P\left(A\right)+P\left(A^{c}B\right)\ge P\left(A\right)\text{since}P\left(A^{c}B\right)\ge 0$$
• (P7) $\begin{array}{c}P(A\cup B)=P\left(A\right)+P\left(A^{c}B\right)=P\left(B\right)+P\left(A{B}^c\right)=P\left(A{B}^c\right)+P\left(AB\right)+P\left(A^{c}B\right)\\ =P\left(A\right)+P\left(B\right)-P\left(AB\right)\end{array}$
  The first three expressions follow from additivity and partitioning of $A\cup B$ as follows (see [link] ).
  $$A\cup B=A\bigvee A^{c}B=B\bigvee A{B}^c=A{B}^c\bigvee AB\bigvee A^{c}B$$
  If we add the first two expressions and subtract the third, we get the last expression. In terms of probability mass, the first expression says the probability in $A\cup B$ is the probability mass in A plus the additional probability mass in the part of B which is not in A . A similar interpretation holds for the second expression. The third is the probability in the common part plus the extra in A and the extra in B . If we add the mass in A and B we have counted the mass in the common part twice. The last expression shows that we correct this by taking away the extra common mass.
• (P8) If $\{B_i:i\in J\}$ is a countable, disjoint class and A is contained in the union, then
  $$A=\underset{i\in J}{\overset{}{\bigvee }}A{B}_i\text{so}\text{that}P\left(A\right)=\sum _{i\in J}P\left(A{B}_i\right)$$
• (P9) Subadditivity . If ${\displaystyle A=\bigcup _{i=1}^{\infty }{A}_i}$ , then ${\displaystyle P\left(A\right)\le \sum _{i=1}^{\infty }P\left(A_i\right)}$ . This follows from countable additivity, property (P6) , and the fact ( Partitions )
  $$A=\bigcup _{i=1}^{\infty }{A}_i=\underset{i=1}{\overset{\infty }{\bigvee }}{B}_i,\text{where}{B}_i=A_i{A}_1^c{A}_2^c\cdots A_{i-1}^c\subset A_i$$
  This includes as a special case the union of a finite number of events.