1. Independent Events

We flip a coin
n times (n 1). For which values of n are the following pairs of events independent?

(a) The first coin flip was heads; the number of all heads was even.
(b) The first coin flip was heads; the number of all heads was more than the number of tails.

(c) The number of heads was even; the number of heads was more than the number of tails.

What I was thinking of doing was listing each case of n (starting with 1) and testing it. But for all I know, this could be long tedious work, especially if it turns out the value of n is quite high. Is there an efficient way to doing this, or should I continue what I was doing by finding the probably of each event for different values of n.

2. I suggest your looking at the case $n=2$.