In my lecture, the equation for the variance of a binomial distribution was derived accordingly:
\begin{align} \sigma^2_x &= Var(y_1+y_2...+y_n) \\\ &= Var\sum^n_{i=1} y_i \\\ &= \sum^n_{i=1}Var (y_i) \\\ &= np(1-p) \end{align}
My question is in regards to the variance being placed inside the summation. Does this only hold true if the variables are independent, or mutually independent? If the condition it needs to satisfy is mutual independence, how so? From my (lack of) understanding about mutually independent events, doesn't there need to be at least 3 possible outcomes of an event?