In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the next two tosses are Tails”.
Well the event $A$ that on the $n$ toss we get two heads followed by two tails is $A = E_n \bigcap E_{n+1} \bigcap E_{n+2}^c \bigcap E_{n+3}^c$
Now is that sufficient or should I be taking a lim sup of the above event? Actually what would it mean to say lim sup $A$ given $A$ as described above?
I don't think you need a lim sup. You just need a countable intersection:
$\bigcap_{n=1}^{\infty}{\left(E_n^c \cup E_{n+1}^c \cup \left(E_n \cap E_{n+1} \cap E_{n+2}^c \cap E_{n+3}^c\right)\right)}$.
Reasoning being that for each $n$, you must have one of those $3$ unioned events occurring: $T$ on $n^{th}$ or $(n+1)^{th}$ toss or the $HHTT$.