3 Scenarios: How many Italian children should Anna expect in "Puzzled Primary School" this year?

Question

"Puzzled Primary School" is a school in London which every year accepts 30 new children in one class, based on random allocation from a large sample. The list has just been allocated for this year.

Anna works at the Italian Consulate in London. As part of her duties, she needs to get every year a list of all schools in England which have enrolled at least one Italian child.

Today she needs to check for “Puzzled Primary School”.

She has also access to past years statistics, and she knows that on average there are 2 Italian children attending Puzzled Primary School every year.

Scenario A

Anna calls "Puzzled Primary School" and speaks with the Admission Officer Lisa. Anna asks Lisa whether there is any Italian children in the new Reception class, and Lisa replies positively.

How many Italian children should Anna expect to be present in the class this year?

Scenario B

Anna is due to call "Puzzled Primary School" to ask whether they have admitted any Italian children this year, but she is busy today and postpones that task to tomorrow.

She soon receives a call from Giulio, who is an Italian father of a child who has been admitted to the school this year. Giulio is quite apprehensive about his son and he has already posted a question on stackexchange (Will the fact that my Italian son is going to attend a primary school change the expected number of Italian children to be present in his class?) to get an estimate of how many Italian classmates his child should expect.

Giulio calls Anna to make sure his child name is recorded in the Italian Consulate register, so that his education will be recognized abroad. He has actually been the first parent to call the Consolate, as parents have 1 year to communicate that via call or letter, and most parents (who are not as apprehensive as Giulio), postpone this action to the last minute. Anna takes his details and records his child attendance to the "Puzzled Primary School".

How many Italian children should Anna expect to be present in the class this year?

Scenario C

Giulio happens to be Anna best friend. He called Anna to let her know his child has been accepted to the school as soon as she knew it, so Anna already knows that before she gets into her office today.

How many Italian children should Anna expect to be present in the class this year?

Answer 1

I think the best way to approach these scenarios is by putting a joint probability distribution on the number of Italian children in class X and A = Anna receiving the answer in the different scenario. We can then use the theorem of Bayes to get the conditional distribution $P(X|A)$ and obtain the answer from there.

Scenario A

Anna get a positive answer A iff $X > 0$. So $E(E|A)=E(X|X>1)$. This is about 2.289.

Scenario B

This one is a bit tricky. Here A = an Italian father is the first parent to call.

You would need to model the probability, that given n Italian fathers for a class, how likely is it that one of them will be the first call. In the extreme case we could assume that they are so concerned, that as long as there are any Italian fathers one of them will call first. Then we are back to Scenario A.

The other extreme would be to assume that every father has the same probability of being the first to call, regardless of nationality. Let's work that one out just a bit using the theorem of Bayes:

$$P(X=k|A)=\frac{P(X=k)P(A|X=k)}{P(A)}$$

Given our stated assumption we have $P(A|X=k)=\frac{k}{30}$. The other terms we get from the binomial distribution for X. Calculating this we get a final result of about 2.9333 for the expected value $E(X|A)$.

The truth is probably in the middle of these two extreme sub-scenarios.

Scenario C

Let us assume that Giulio will always tell Anna if his kid is going into Puzzled Primary School. If the number of Italian children in the population is sufficiently large, then the probability of the child of Giulio being in the class is approximately proportional to the number of Italian children in the class, or

$$P(A|X=k)=\alpha\frac{k}{30}$$

Fortunately $P(A)$ also scales linearly with $\alpha$, so that it will cancel out in the equation above. We again obtain 2.9333 as the expected number of Italian children.