Q:

Find the probability that when n balls are extracted, between a and b of them are white

An urn contains white and black balls with $p_w = p$ and $p_b = 1-p$. If we made $n$ extractions with replacement what is the probability that the number of white balls extracted is between $a$ and $b$?

I thought this:

$P=P(b) - P(a)$
$P(b) =
   \binom{n}{b}\cdot\left(p\right)^{b}\cdot\left(1-p\right)^{n-b}$
$P(a)
   = \binom{n}{a}\cdot\left(p\right)^{a}\cdot\left(1-p\right)^{n-a}$

Am I doing it right?

A:

You need to be slightly careful with this situation, since we are looking
for multiple sample points, not just the sample points that our extraction
gives us $b$ white balls and our extraction gives us $a$ white balls. 
What
we are looking for is
$P(a \leq X \leq b)$, where $X \sim Binom(n,p_w)$ is the random variable
representing the number of white balls picked. 
By the cumulative mass
function of the binomial distribution,
$P(a \leq X \leq b) = \sum_{i = a}^b \binom{n}{i} p^i (1-p)^{n-i}.$
You can simplify this using  the binomial theorem.

