In a village with 2000 people, 100 people suffer from Alzheimer's disease. On a certain day, 40 people are admitted to a hospital.
Calculate the probability that between $15$ and $25$ people (inclusive) of the patients admitted have Alzheimer's disease.
My attempt:
Let $X$ be the patients admitted who have Alzheimer's disease. $$P(15 \leq x \leq 25) = P(X=25) - P(X=15) = \frac{\binom{100}{25} \binom{1900}{15}}{\binom{2000}{40}}-\frac{\binom{100}{15} \binom{1900}{25}}{\binom{2000}{40}} $$
What is the expected number of patients who have Alzheimer's disease?
$$ E(X) = 40 \frac{100}{2000} =2$$
Is this right?
The probability is wrong. Note that $$P(15\leq X \leq 25) = P(X \leq 25) - P(X \leq 15) = \sum_{k=15}^{25} P(X=k).$$
The expected number is correct.