Suppose that five coins are each tossed until the first head is obtained on each coin and where each coin has a probability $\theta$ of producing a head. If you are told that the total number of tails observed is $10$, then determine the expected number of tails observed on the first coin.
I know the solution is as follows:
$$10 = E(X_1 + \cdots + X_5 | X_1 + \cdots + X_5 = 10)$$ $$= \sum E(X_i | X_1 + \cdots + X_5 = 10)$$ $$= 5 \times E(X_1 | X_1 + \cdots + X5 = 10)$$
$$E(X_1 | X_1 + \cdots + X_5 = 10) = 2$$
My questions are:
- why the conditional expectation equals to the total number of tails? Does that mean conditional expectation represent the times that something happens?
- why each coins has the same conditional expectation(expected number)?