Thanks for your explanation. I did not get it in the first read, and was too lazy to think, until saw your comment.
Just want to point out, that one is actually doing the experiment with a biased coin, then one must ignore all pairs.
e.g in case a coin which is heavily biased, say .9 H and .1 T. One should start with ignoring all the HH pairs, and start only at odd index. Lest, one picks a value like HHHHT (in the case the 2nd HH pair was not skipped, instead they greedily picked up the first HT, which will make the experiment HT biased).
Just want to point out, that one is actually doing the experiment with a biased coin, then one must ignore all pairs.
e.g in case a coin which is heavily biased, say .9 H and .1 T. One should start with ignoring all the HH pairs, and start only at odd index. Lest, one picks a value like HHHHT (in the case the 2nd HH pair was not skipped, instead they greedily picked up the first HT, which will make the experiment HT biased).