I'm confused about how to reconcile the probability of independent events not having anything to do with prior history, but sequences of events do (seemingly) take into account prior history. This question asks a similar question: Probability of independent events given the history. However, having read that, I found I had a very specific confusion about the seeming contradiction between two formulas for probabilities that seem equal to me, but will produce different results based on our understanding of P of sequences versus P of independent events:
(A) P(HHHHH) = 0.03125
(B) P(H | HHHH) = 0.5
Can anyone explain how the left side of both equations, P(HHHHH) and P(H | HHHH) are different.
And does anything change if we shift from a frequentist to bayesian perspective?