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? This is a question stemming from a real-life situation, for which I have been genuinely puzzled about its answer.
My son is due to start primary school in London. As we are Italian, I was curious to know how many Italian children are already attending the school. I asked this to the Admission Officer while applying, and she told me they have on average 2 Italian children per class (of 30).
I am now at the point in time where I know that my child has been accepted, but I have no other information about the other children.
Admission criteria are based on distance, but for the purpose of this question, I believe we could assume it's based on random allocation from a large sample of applicants.
How many Italian children are expected to be in my son's class? Will it be closer to 2 or 3?
 A: Here's my thoughts on how to approach this: 
Let the random variable $S_n$ denote the number of Italian children in a class that is currently of size $n$. Let $X$ be the indicator for a new child's being Italian. Suppose that we add child $X$ to this class. Then the expected number of Italian children in this augmented class of size $n+1$ is $\mathbb E(S_n + X) = \mathbb E(S_n) + \mathbb E(X) = \mathbb E(S_n) + \mathbb P(X = 1)$. Note that independence doesn't matter here since we're only using the linearity of expectation. If child $X$ is known to be Italian then $X = 1$ with probability 1 so we have increased the expected value by 1.
A: As always you need to consider a probabilistic model that describes how the school distributes children among classes. Possibilities:


*

*The school takes care that all classes have the same number of foreign nationals.

*The school even tries to make certain that each nationality is represented roughly the same in every class.

*The school doesn't consider nationality at all and just distributes randomly or based on other criteria.


All of these are reasonable. Given strategy 2 the answer to your question is no. When they use strategy 3, the expectation will be close to 3, but a bit smaller. That is because your son takes up a "slot", and you have one less chance for a random Italian. 
When the school uses strategy 1 the expectation also goes up; how much depends on the number of foreign nationals per class. 
Without knowing your school there is no way to answer this more perfectly. If you have just one class per year and the admission criteria are as described the answer would be the same as for 3 above.
Calculating for 3 in detail:
$$E(X) = 1 + E(B(29, 2/30)) = 1 + 1.9333 = 2.9333.$$
X is the number of Italian children in the class. The 1 comes from the known child, the 29 are the rest of the class and 2/30 is the probability for an unknown kid being Italian given what the school says. B is the binomial distribution.
Note that starting with $E(X|X\geq1)$ does not give the proper answer, as knowing that a specific child is Italian violates the exchangeability assumed by the binomial distribution. Compare this with the boy or girl paradox, where it makes a difference whether you know that one child is a girl vs. knowing that the older child is a girl.
A: Another way to look a this is at the level of individual children. Assuming that 30 children drawn randomly from a population (which you've indicated we can), we can work backward to the rough probability of an Italian child being drawn from this population: $2/30$ = $1/15$.
Given that we know that one of the 30 is Italian, we only have to compute the probability for the remaining children:
$$29 \cdot 1/15 = 29/15 = 1.933\ldots$$
So, knowing that your child is Italian changes the expected number of Italian children in the class to approximately 2.933, which is much closer to 3 than 2.
A: Based on the Admission Office info, the number of Italian children follows binomial $\mathrm{Binom}(30, 2/30)$, assuming independence. Now you know in your class, there is at least one Italian child, so the expectation becomes $\mathbb{E}(X|X\geq1)$. For $X\sim \mathrm{Binom}(30, 2/30)$, this evaluates to $2.28$ (if I get my calculation right).

Edit. Evaluation of the expectation:
$$E[X|X\geq1]=\sum_{i=0}^{30}iP(X=i|X\geq1)=\sum_0^{30}i\cdot \frac{P(X=i, X\geq1)}{P(X\geq1)}=\sum_1^{30}i\cdot \frac{P(i)}{1-P(0)}$$
(note the change in summation lower bound at last step)
A: No.
Your knowledge of the impending events changes nothing about the school's typical experience.
