Second Toss is Head
Suppose we flip a fair coin \(n\) times (independently). What is the expected number of heads given that the second toss is a head?
Answer
Let $S = \sum_{i=1}^n I_i$ be the total number of heads, where \(I_i\) is the indicator RV that the \(i\)-th toss is a head. Using independence of flips, we then have:
\[
\mathbb{E}[S \mid I_2=1]
=
1 + \sum_{i \ne 2} \mathbb{E}[I_i \mid I_2=1]
=
1 + \frac{n-1}{2}
=
\frac{n+1}{2}.
\]