Having some trouble with these two...
Suppose a health insurance company can resolve 60% of claims using a computerised system, the remaining needing work by humans. On a particular day, 10 claims arrived, assuming claims are independent, what is the probability that:
Q2.1) Either 3 or 4 (inclusive) claims require work by a human?
Q2.2) No more than 9 claims require work by a human?
I have identified that:
$n=10, p=0.6, q=0.4$
How would I go about these questions? Any help would be appreciated!
For 2.1, we can separately compute $3$ or $4$ claims needing human intervention.
For $3$ claims, we have $\displaystyle \binom{10}{3}(0.6)^7(0.4)^3$, and for $4$ claims, we have $\displaystyle \binom{10}{4}(0.6)^6(0.4)^4$. We simply add these probabilities up.
For 2.2, rather than adding up everything from $0$ to $9$, we can just compute $10$, and do $1-$that probability. For $10$, we have $(0.4)^{10}$.