The events are all independent from each other so your friend is right, the probability does stay the same.
I had an argument with a friend recently. I know for sure that, say, flipping a coin one time, has a 0.5 probability of being either result. So does my friend. However, he is convinced that with further flips, the probability remains constant, as opposed to being for getting the same result.
What is the simplest possible way I can explain this to him?
The question you originally posted is obviously not the question you have in mind. Since we are not mind readers, you have to accept the reply given to the question you actually posted, not the question you wanted to post.
There is a difference between the question "What is the probability of getting a head on the nth toss" and the question "What is the probability of getting n heads in a row?"
Your original post implied the former not the latter. And the former is what got answered.
Assuming that the latter question is in fact the actual question you wanted to ask, I'd first consider the simple case of tossing two heads in a row. The possible outcomes are:
TT, TH, HT, HH.
All outcomes are equally likely. Therefore the probablity of getting HH is 1/4 (= 0.5^2).
Then consider three tosses etc.