<s>[INST] Probability of number of events where $p_{n+1} = \frac{1}{5}p_n$ For $ n \ge 0, p_{n+1} = \frac{1}{5}p_n $. Where $p_n$  is the probability of an envent occuring n times in a period.
What is the probability that more than one event occurs?
 [/INST] $1= \sum\limits_{n=0}^{\infty}(\frac{1}{5})^{n}p_0 $
$(\frac{1}{5})(1) = (\frac{1}{5})\sum\limits_{n=0}^{\infty}(\frac{1}{5})^{n}p_0 $
$1 - (\frac{1}{5})(1) = p_0 (\frac{1}{5})^{0}$
$ \frac{4}{5} = p_0$
using the formula in the question,
$p_1 = \frac{1}{5}(\frac{4}{5})$
$p_1 = \frac{4}{25} $
$P(n\ge 2) = 1 - P(n=0) - P(n=1)$
$P(n\ge 2) = 1 - \frac{4}{5} - \frac{4}{25}$
$P(n\ge 2) = 0.04$
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