Independent Events of n coin tosses If the probability of heads when tossing a coin is p. This coin is tossed n times. If H is the event of getting a Head on the first toss and $F_k $ is the event exactly k Heads are tossed, for which pairs of integers (n,k) are H and $F_k $ independent,
I can't really see how these events will ever be independent? Given the first toss is a head it will always affect the event of an exact number of total heads? But the question suggests finding these pairs of integers? any help?

By definition two events A and B are independent if $P(A\cap B)=P(A)P(B).$ 
In this case $P(A)=P(H)=$Probability of getting heads on first try and 
$P(B)=P(F_k)=$Probability of getting exactly k heads. We know that $P(A)=p$ and we also know that $P(B)={n \choose k}p^k(1-p)^{n-k}$ so $P(A)P(B)=p{n \choose k}p^k(1-p)^{n-k}$. 
$P(A\cap B)$ is the probability of getting heads on the first toss and then exactly $k-1$ more heads so $P(A\cap B)=p{n-1 \choose k-1}p^{k-1}(1-p)^{n-k}$. 
Setting the two equal to each other yields ${(n-1)! \over (k-1)!(n-k)!} p^k(1-p)^{n-k}={(n)! \over (k)!(n-k)!}p^{k+1}(1-p)^{n-k}$ which upon simplification yields $k=np$ so for instance if $p={1 \over 2}$ and $n=2$ then $H$ and $F_k$ are independent.
