Let $\xi_1, \xi_2, \ldots$ be a sequence of random variables that describe a sequence of coin tosses:
$\xi_n=\begin{cases}+1, & \mathrm{head}\\-1, & \mathrm{tail}\end{cases}$
Let $\mathcal{F}_n$ be the $\sigma-$Algebra generated by $\xi_1, \xi_2, \ldots, \xi_n$
I want to find the smallest $n$ so that the event is in $\mathcal{F_n}$ if such a $n \in \mathbb{N}$ exists.
a) A=Head appears for the first time after at most $10$ times tail.
b) B=Head appears at least once in the whole sequence.
c) C=The first $100$ tosses have the same result.
d) D=Head and tail don't appear more than twice respectively in the first 10 tosses.
My attempt
a) $11$
b) such a $n \in \mathbb{N}$ doesn't exist
c) $100$
d) This event is impossible, so it is in $\mathcal{F}_0=\{\emptyset,\Omega\}$