An insurance company found that 25% of all insurance policies

Step 1
Given,
An insurance company found that all insurance policies terminated before their maturity date.
The Probability of insurance terminated before the maturity date is 25%
${\displaystyle q=0.25}$
Then, the Probability of insurance not terminated before the maturity date is, ${\displaystyle p=1-0.25=0.75}$
Assume that ${\displaystyle n=10}$ policies are randomly selected from the company’s policy database.
Let success be the insurance not terminated before the maturity date the required calculation can be defined by the formula,
${\displaystyle P(X=x)=n{C}_x{p}^x{q}^{n-x}}$
n number o samples
x number of success
p probability of success
q probability of failure
Step 2
to find the required probability:
${\displaystyle P(X\le 8)=\sum _{x=0}^{8}10C_x{\left(0.75\right)}^x{\left(0.25\right)}^{10-x}}$
${\displaystyle =1-\sum _{x=9}^{10}10C_x{\left(0.75\right)}^x{\left(0.25\right)}^{10-x}}$
${\displaystyle =1-[10C_9{\left(0.75\right)}^9{\left(0.25\right)}^{10-9}+10C_{10}{\left(0.25\right)}^{10-10}]}$
${\displaystyle =1-[0.1877+0.0563]}$
${\displaystyle =1-\left[0.2440\right]}$
${\displaystyle =0.7560}$
Therefore the probability that at most eight policies are not terminated before maturity is 0.7560 or 75.60%