Not having read the paper to which to parent referred, but I might be able to shed some light on this... Let us describe the state of the previous two tosses a

Not having read the paper to which to parent referred, but I might be able to shed some light on this... Let us describe the state of the previous two tosses as XX, with X being either H or T. If --(X)--> describes the next toss, we can describe the transition between states as follows: HH --(H)--> HH HH --(T)--> HT and HT --(H)--> TH HT --(T)--> TT etc. If we are waiting for HHT, starting from state HH HH --(H)--> HH : or, we get back to state HH HH --(T)--> "HHT" : HHT is found and we stop searching as opposed to waiting for HHH HH --(H)--> "HHH" : HHH is found, and we stop HH --(T)--> HT : and we are in a new state which requires at least two more --(H)--> in sequence before we return to state HH. I hope this helps. EDIT: reformatting & typo