Calculating theoretical probability

Question

I have the following question in one of my tutorial.

Background: A manufacturing company developed 40000 new drugs and they need to be tested. Question The QA checks on the previous batches of drugs found that — it is four times more likely that a drug is able to produce a better result than not. If we take a sample of ten drugs, we need to find the theoretical probability that at most 3 drugs are not able to do a satisfactory job.

I think we need to use the cumulative probability distribution $F(3) = P(X\lt 3).$ However not sure how to calculate it. Any guidance is helpful.

Answer 1

If you let $G$ be the event that a drug produces better results and you let $B$ be the event that a drug produces not as good of results, then the first step in your question is to solve $4P(B)=P(G)$ with the condition that $P(B) + P(G)=1$. From this, we get $P(B)=\frac{1}{5}$ and $P(G)=\frac{4}{5}$.

Accordingly, if you let $X$ be the event that out of 10 trials, you pick $x$ amount of drugs that do not produce as good of results as the other drug, then $X$ is a random variable with a Binomial Distribution, where $n=10$ and $p=1/5$.

So, $P(X=x)={10 \choose x}\frac{1}{5}^{x}\frac{4}{5}^{10-x}$, for $0 \leq x \leq 10$.

All that is left is to compute $P(X \leq 3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=\sum_{i=0}^{3}{10 \choose i}\frac{1}{5}^{i}\frac{4}{5}^{10-i}$

Answer 2

The fact that it is a sample means that the event of each being good can be assumed to be independent from each other. This is enough information to calculate the odds of any particular pattern of good vs bad. For example GBGGBGGGBG will turn up with probability $0.8 * 0.2 * 0.8 * 0.8 * 0.2 * 0.8 * 0.8 * 0.8 * 0.2 * 0.8 = 0.2^3 * 0.8^7$

Now you just have to sum up the probabilities of seeing all patterns that satisfy your rule.