Independence tells us how to compute the probability of a sequence like HT or TH:

  P(HT) = P(H)P(T) = p(1 - p)

But the question I am addressing is not just "what is the probability of HT?" It is "given that the two flips are different, what is the probability that the order was HT rather than TH?"

That is a conditional probability:

  P(HT | HT or TH)

That wasn’t what he was trying to prove, but the proof could be done without conditionals like this:

If: p(H)p(T) = p(T)p(H)

And: p(H)p(T) + p(T)p(H) = 1

Then: p(H)p(T) = p(T)p(H) = 0.5

Thats how i noodled thru it internally