probability of null set

probability of null set

$P\left(\varnothing \right)=0$ where $\varnothing$ is the null event (the event containing no outcomes whatsoever).

This in turn implies that the property contained in Axiom 3 is valid for a finite collection of disjoint events.

\begin{proof} First consider the infinite collection $A_1=\varnothing ,A_2=\varnothing ,A_3=\varnothing ,\dots .$ Since $\varnothing \cap \varnothing =\varnothing$, the events in this collection are disjoint and $\cup A_i=\varnothing$. The third axiom then gives

$$P\left(\varnothing \right)=\sum P\left(\varnothing \right)$$

This can happen only if $P\left(\varnothing \right)=0$.

Now suppose that $A_1,A_2,\dots ,A_k$ are disjoint events, and append to these the infinite collection $A_{k+1}=\varnothing$, $A_{k+2}=\varnothing$, $A_{k+3}=\varnothing$, … . Again invoking the third axiom,

$$\begin{array}{rl}P\left(\bigcup _{i=1}^k{A}_i\right)& =P\left(\bigcup _{i=1}^{\infty }{A}_i\right)\\ & =\sum _{i=1}^{\infty }P\left(A_i\right)\\ & =\sum _{i=1}^{k}P\left(A_i\right)\end{array}$$

as desired. \end{proof}

• EX 2.11 toss a thumbtack
• EX 2.12 test batteries