I am handed a fair coin. I understand that the probability of H/T is 0.5.
I flip and observe H. I expect that on my 2nd toss, the probability of H is still 0.5
I flip and observe H. Again I expect that upon my 3rd toss, p(H) = 0.5.
Why is it that I should expect p(H) = 0.5 on my 3rd trial? If I use the binomial distribution, I know that observing 3 successes in 3 trials is actually p=0.125. I did not set out to flip the coin 3 times as a set, yet I've done so.
What am I missing or forgetting?