In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative integer up to $5$.
\[
p_{n+1}=\frac{1}{2} p_{n}
\]
where $p_{n}$ denotes the probability that a health insurance policy holder files $n$ claims during this period. The analyst assumes that no individual files more than $5$ claims in this period. Under these assumptions, what is the probability that a policy holder files more than two claims in this period?
- $7 / 31$
- $29 / 125$
- $1 / 3$
- $13 / 125$
- None of the above