Probability question - Bernoulli trials vs Binomial distribution I am handed a fair coin. I understand that the probability of H/T is 0.5.
Trial 1.
I flip and observe H. I expect that on my 2nd toss, the probability of H is still 0.5
Trial 2.
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?

If it is a fair coin why would suddenly the probability of the heads change? Just think of physics of the phenomenon, it can’t be possibly affected by what happened in previous coin tosses. A coin can't remember what was a previous toss, and can't know when did you start observing its tosses.
Besides, the binomial distribution assumes that the tosses are independent. If you make coin tosses dependent then you can’t use binomial distribution. No eating the cake and having it too
