Problem:
There is a machine which runs a test for which a spoilt orange gets approved with a probability of 0.15 and a normal orange(in good condition) gets approved with a probability of 0.95.
Now we need to create a new test which the machine runs $n$ independent trials and requires at least $m$ independent successes. I need to find the values of $m$ and $n$ such that the spoilt orange gets approved with a probability of at most 0.1 and a normal orange gets approved with a probability of at least 0.99.
Note that m and n $\in$ {1,2,3,4,5}.
Understanding:
I have tried using the binomial distribution formula
$$ P(X = m) = {n \choose m}p^m(1 - p)^{n-m}. $$
with pair values of n and m within the set mentioned where n $\geq$ m to find the suitable pairs by trial and error. However, when it came to finding the pair values for normal orange gets approved with p = 0.99, there are no pairs as the maximum value I got was 0.95. This is when use p = 0.95 in the formula.
I'm not sure if this is right and some guidance would be very much appreciated.
Hint: The probability that an orange gets approved is $\ P(X\ge m)=\sum_\limits{i=m}^n {n\choose i}p^i(1-p)^{n-i}\ $. For $\ m=1\ $ this is $\ 1-(1-p)^n\ $, which is $\ 0.999997\ $ for $\ n=5\ $ and $\ p=0.95\ $, so there are certainly some values for $\ m\ $ and $\ n\ $ for which a normal orange will be approved with probability at least $\ 0.99\ $. For that value of $\ m\ $, of course, spoilt oranges will get approved with probability $\ 1-(1-0.15)^5=0.55\ $, so this value of $\ m\ $ won't satisfy your other criterion.
If you increase $\ m\ $, however, then the probability of either type of orange's being approved will decrease. You need to make it sufficiently large for the probability of a spoilt orange's being approved drops below $\ 0.1\ $, but not so large that the probability of a good orange's being approved drops below $\ 0.99\ $. This is possible with $\ n=5\ $ and also for at least one other value of $\ n\ $ smaller than $5$. I'll leave you to discover the possible values of $\ m\ $ and $\ n\ $ for yourself.