# Reciprocal of a binomial random variable

A production line produces faulty items independently at random with probability p.

(a) Let X be the number of faulty items in a batch of 10. What is the distribution of X, and what is P(X = k)?

(b) Let Y be the number of items you have to check to find 2 faulty ones. What is the distribution of Y, and what is P(Y = n)?

(c) Give values of k and n for which P(X ≥ k) = P(Y ≤ n).

P(X) = 10CX . p^X . (1-p)^(10-X)

a) P(X=k) = 10Ck . p^k . (1-p)^(10-k)

b) Yp = 2 -> Y = 2/p = 2/(X/10) = 20/X

I don't know how to proceed from here. I am tempted to find an inverse distribution function. But I don't think it is appropriate for discrete function and more than that I don't think the question is asking for a complex solution.

You can correct if my solution to part a is incorrect, ie if the variable X is not binomial distributed.

• @kjetilbhalvorsen I added the tag. I need to see some similar examples atleast. Not wiki page Commented Feb 13, 2020 at 17:24
• @DomJo it means read the wiki of self-study tag Commented Feb 13, 2020 at 17:32
• (b) makes little sense because $X$ and $Y$ are counts. That suggests reconsidering what question (2) is asking.
– whuber
Commented Feb 13, 2020 at 17:59
• Have you perchance been learning about applications of the negative binomial distribution?
– whuber
Commented Feb 13, 2020 at 18:13

$$P(n) = {^{n-1}}C_{2-1}.p^{2}.(1-p)^{n-2}$$