Comment by susam

And for people who like equations, here is my attempt at explaining it.

Assume each flip is independent and the bias remains same in each flip.

Let

  P(H) = p,
  P(T) = 1 - p.

Then

  P(HH) = p^2,
  P(HT) = p(1 - p),
  P(TH) = (1 - p)p,
  P(TT) = (1 - p)^2.

Therefore

  P(HT or TH) = 2p(1 - p).

Now calculate

  P(HT | HT or TH) = p(1 - p) / (2p(1 - p)) = 1/2,
  P(TH | HT or TH) = (1 - p)p / (2p(1 - p)) = 1/2.