I'm stuck with a part of a homework problem and need some clarification. Let $\Omega$ be the sample space for flipping $n$ fair coins, i.e. the set of all $n$-tuples of $E$ and $N$, eagle and number, the $\sigma$-algebra $\cal{F}$ of all subsets of $\Omega$ and $P(A)=\frac{|A|}{2^n}$.
Let $X_k$ denote the number of the eagles in the first $k$ flips, i.e. if $\omega=(x_1,...,x_n)$, then $X_k(\omega)=|\{1\leq i\leq k:x_i=E\}|$. Describe $\cal{F}_{X_k}$. How many elements does it have? Give an example of an element in $\Omega$, but not in $\cal{F}_{X_k}$. Show that that $E[X_n |\cal{F}_{X_k}]= X_k+\frac{n-k}{2}$
Wouldn't $\cal{F}_{X_k}$ be a set of all $k$-tuples, hence $2^k$ elements? But then no element of $\cal{F}_{X_k}$ would be in $\Omega$, so that seems not correct. But what is $\cal{F}_{X_k}$ then? Or is this just a trick question?
For the second part, this is the expectation of flipping $n$ coins after the result of the first $k$ coins is known; then we are left with $\frac{n-k}{2}$ eagles out of $n-k$ further coins. But how do I prove this formally?
Yours, Marie