I have the following...
$$\sum_{i=1}^n x_i - p\sum_{i=1}^n x_i = p\sum_{i=1}^n (1-x_i)$$
And I'm told it can be re-written as so to isolate for $p$...
$$p = \dfrac{1}{n}\sum_{i=1}^n x_i$$
It is unclear to me why that is true. I tried to isolate $p$ but only got so far...
$$p = \frac{\sum_{i=1}^n x_i}{\sum_{i=1}^n x_i + \sum_{i=1}^n (1-x_i)}$$
Can someone explain to me what summation property is used and how it is used to isolate for $p$?
Note that the denominator in your expression for $p$ can be written as \begin{align*} \sum_{i=1}^n x_i + \sum_{i=1}^n (1-x_i) &= \sum_{i=1}^n x_i + \sum_{i=1}^n 1 - \sum_{i=1}^n x_i\\ &= \sum_{i=1}^n x_i + n - \sum_{i=1}^n x_i = n. \end{align*} Thus, your expression becomes $$ p = \frac{\sum_{i=1}^n x_i}{\sum_{i=1}^n x_i + \sum_{i=1}^n (1-x_i)} = \frac{\sum_{i=1}^n x_i}{n}. $$