Question

Consider testing batteries coming off an assembly line one by one until one having a voltage within prescribed limits is found. The simple events are E1= {S}, E2= {FS}, E3= {FFS} and so on .Suppose the probability of any particular battery being satisfactory is 0.99. All Ei are disjoints. Write down complete sample space and prove that P (sample space) =1.

Answer 1

It's clear enough that

$\mathbb P(E_i)=\mathbb P(i=j)=\begin{cases}0.99&\text{for }j=1\\0.01\times0.99&\text{for }j=2\\0.01^2\times0.99&\text{for }j=3\\\vdots\end{cases}$

Because each of the $E_i$ are disjoint, it follows that

$\displaystyle\mathbb P(\text{sample space})=\mathbb P(E_1\cup E_2\cup\cdots)=\mathbb P\left(\bigcup_{i=1}^\infty E_i\right)=\sum_{i=1}^\infty\mathbb P(E_i)$
$=0.99+0.01\times0.99+0.01^2\times0.99+\cdots$
$=0.99(1+0.01+0.01^2+\cdots)$
$=0.99\displaystyle\sum_{n=0}^\infty 0.01^n$
$=0.99\times\dfrac1{1-0.01}$
$=\dfrac{0.99}{0.99}=1$