A factory has $2$ firms producing the same type of product. The numbers of errors per product

produced by firm A and firm B follow Poisson distributions with means of $0\mathrm{.}1$ and $0\mathrm{.}2$ respectively.

Furthermore, errors occur independently between products regardless the producing firms.

$($a$)$ Suppose that the proportion of products produced by firm A is $0\mathrm{.}25.$ In a random sample of $15$

products produced by this factory, find the probability that there are more than $12$ products

that have exactly $3$ errors.

$($b$)$ In a random sample of $100$ products produced by firm A$,$ find the probability that there are

from $60$ to $95$ products that have at least $1$ error.