This one is a solved example from Feller.
Spores of the Fungus are produced in chains of eight.The chain may break into several parts into projectiles containing 1 to 8 pores. Find the expected no. of doublets if all links have probability $p$ to break.
If we assume $S$ to be the random variable which assumes the no. of doublets formed, then we need to evaluate $E[S]$. The maximum value $S$ can assume is $4$. However, Feller has expressed $S=S_1+S_2+S_3+S_4+S_5+S_6+S_7$ as there are $7$ different ways a doublet can be formed where $S_i \in \{0,1\}$. I'm unable to comprehend the equality of $S$ with $\sum S_i$ as they have different range with $S \in \{0,4\}$ and $\sum S_i \in \{0,1,2,3,4,5,6,7\}$.
Can someone explain the rationale behind it?
$\sum S_i$ has range $\{0,1,2,3,4\}$ not $\{0,1,2,3,4,5,6,7\}$. This is because the $S_i$ are dependent on each other. In particular,
\begin{eqnarray*} S_1 = 1 &\implies& S_2 = 0 \\ S_2 = 1 &\implies& S_1 = S_3 = 0 \\ S_3 = 1 &\implies& S_2 = S_4 = 0 \\ S_4 = 1 &\implies& S_3 = S_5 = 0 \\ S_5 = 1 &\implies& S_4 = S_6 = 0 \\ S_6 = 1 &\implies& S_5 = S_7 = 0 \\ S_7 = 1 &\implies& S_6 = 0. \end{eqnarray*}
$\sum S_i$ attains its maximum of $4$ when $(S_1,\ldots,S_7) = (1,0,1,0,1,0,1)$.