What is the variance and expectation of the number of people wearing black socks?

Question

Suzy, knowing people own cats with probability p and wear black socks (by total coincidence!) with probability p, interviews people on the street up to and including the first cat owner she meets and then asks them if they’re wearing black socks. What is the variance and expectation of the number of people wearing black socks?

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Okay, I've got some questions here. Is saying that "interviews people on the street up to and including the first cat owner she meets" means that we have a geometric distribution here?

I'm a bit confused because after meeting everybody and stopping at the first cat owner shes will question them on the colour of their socks (black) like they will hypothetically all wait for her to finish the first task.

Answer 1

Okay, I've got some questions here. Is saying that "interviews people on the street up to and including the first cat owner she meets" means that we have a geometric distribution here?

Yes.   Let $X$ be the count of people she asks.   Then $X\sim\mathcal{Geo}_1(p)$.

I'm a bit confused because after meeting everybody and stopping at the first cat owner shes will question them on the colour of their socks (black) like they will hypothetically all wait for her to finish the first task.

She asks two questions of every person she interviews, "do you wear black socks?" and "do you own a cat?", keeping tally of the answers.   She stops asking after getting the first positive answer to the second question.

Let $Y$ be the count of people wearing black socks among the people asked.   The conditional expectation for this count for a given $X$ will be Binomially distributed.   $Y\mid X\sim\mathcal{Bin}(X, p)$

What is the variance and expectation of the number of people wearing black socks?

Your task is to evaluate $\mathsf E(Y)$ and $\mathsf {Var}(Y)$ from these distributions.   Use the Laws of Total Expectation and Total Variance.

Answer 2

Presumably, owning a cat and wearing black socks are to be treated as independent events (although, if cat owners prefer to wear socks that don't show their cat's hair as much, that might not be the case). The number $Y$ of people interviewed is indeed a geometric random variable (in the version that has support $\{1,2,3,\ldots\}$, i.e. the number of trials up to and including the first success). If $X$ is the number of people interviewed wearing black socks, then the conditional distribution of $X$ given $Y$ is binomial with parameters $Y$ and $p$.

My interpretation of the question is that she asks each person about socks immediately after asking that person about cats. That's certainly more practical than having to keep everybody waiting until she meets a cat-owner, but (unless the interview subjects get so annoyed at waiting that they don't answer correctly) the resulting distribution would be the same.