There's a problem on expectation for a discrete random variable that involves people being selected as happy/unhappy. I am stumped, I thought I might reach out for help on the theory.
The problem is
"Consider a group of $n >= 4$ people, numbered from 1 to $n$. For each pair $(i, j)$ with $i =/= j$ person $i$ and person $j$ are friends, with probability $p$. Friendships are independent for different pairs. These $n$ people are seated around a round table. For convenience, assume that the chairs are numbered from to $1$ to $n$, clockwise, with $n$ located next to 1, and that person $i$ seated in chair $i$. In particular, person 1 and person $n$ are seated next to each other.
If a person is friends with both people sitting next to him/her, we say this person is happy. Let $H$ be the total number of happy people. Express your answers in terms of $p$ and/or $n$"
It says Lets $I_i$, be a random variable indicating whether the person seated in chair i is happy or not ($I_i$ = 1 if person $i$ is happy and 0 otherwise). Find $E[I_i]$ for $i$ = 1,2,...,n
I modeled it out in a circumstance where $n$ = 5. I'm stumped by the question of how to handle p. Is p different for each pair? I know the $E[I_a]$ for an indicator variable is $p$, but that doesn't seem to apply because $p$ seems specific to each pair.
It has a couple other questions, like $E[H]$ and $E[I^2 sub k]$ for any $k$ in set ${1,2,..n}$ but I'm so stuck on the first part that these seem very far away.
You can assume that $p$ is the same for every pair $(i,j)$; if the probability depended on which people were involved, you'd see a subscript on $p$, as in "person $i$ and person $j$ are friends with probability $p_{i,j}$".
To calculate $E(I_i)$: This is the probability that person $i$ is happy, which by definition is the probability that person $i$ is friends with the person to the left, and person $i$ is friends with the person to the right. These two events are independent (since friendships are independent for different pairs), and each event has probability $p$, so the probability of their intersection ("and") is $p^2$, the product of the individual probabilities. Therefore $E(I_i)=p^2$ for every $i$.