Let X be the number of accidents in a factory per week. , It has the pmf
f(x) = 1 / (x+1)(x+2) ; x=0,1,2,...
Find the conditional probability of x > or = 4, given that x >or = 1
$\displaystyle f(x) = \frac{1}{(x+1)(x+2)} = \frac{1}{x+1} - \frac{1}{x+2}$.
Therefore $\displaystyle \Pr(x \geq 4) = \sum_{x=4}^{\infty} \left( \frac{1}{x+1} - \frac{1}{x+2} \right) = \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + ...... = \frac{1}{5}$.
(This is an example of a telescoping series)
$\displaystyle \Pr(x \geq 1)$ is got in a similar way ....