# Does conditional expectation represent the times that something happens?

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:

$$10 = E(X_1 + \cdots + X_5 | X_1 + \cdots + X_5 = 10)$$ $$= \sum E(X_i | X_1 + \cdots + X_5 = 10)$$ $$= 5 \times E(X_1 | X_1 + \cdots + X5 = 10)$$

hence,

$$E(X_1 | X_1 + \cdots + X_5 = 10) = 2$$

My questions are:

1. why the conditional expectation equals to the total number of tails? Does that mean conditional expectation represent the times that something happens?
2. why each coins has the same conditional expectation(expected number)?
• what are the values of the random variables H (head) and T (tails)? it is not clear in your question – Nikos M. Aug 16 '15 at 15:22
• You are estimating the sum of the expected number of tails given that you know you got a sum of 10. That is all you know; so the best estimator will naturally be $10$. Since every coin is assumed to be equal in all respects you can first use linearity of expectations, and later bring the $5x$ in front. In this tautonomic example of conditionality, i.e. $E(\sum X|\sum X=10)=\sum X$, your condition is your estimate; typically, you estimate a parameter ($p$) given observed outomes, and then the expectation is different from the observation. – Antoni Parellada Aug 18 '15 at 13:30

1. "each coin has a probability $\theta$ of producing a head", this means that $P(X_i = H)=\theta$, $P(X_i = T)=1-\theta$
2. Expectation on $X_i$ is $E[X_i] = \theta \times H + (1-\theta) \times T$, where $H$, $T$ are random variables which take some values
3. But here i lose you, what are the random variables $H$ and $T$ values?