Coin tossing conundrum

This is from The Mathematics of Various Entertaining Subjects: Research in Recreational Math.

It left me so puzzled it took me a while to get my head around it.

It’s game – Flipping Fun – and the idea is that participants pick a set of three coin tosses (eg THH, HHH, THT and so on) and the winner is the person who has picked the first sequence that comes up.

The mind bending part of this is:

  • The coin is fair so the odds of it turning up as heads or tails on any toss are the same (1/2).
  • Thus any sequence of a given length is equally likely – i.e. if the coin is tossed ten times then HHHHHHHHHH is as likely as HTHTHTHTHT or any other sequence.
  • Despite both of the above facts some sequences of three tosses are much more likely to win than others.

To illustrate this point, think of HHT against TTT. Here HHT is much more likely to win – because once two heads have been tossed (odds 1/4) then it is only a question of waiting for a tail to come up, as tossing a further head keeps the HH sequence alive.

So, for instance, in five tosses the odds of HHT winning are 1/4 (8/32), but the odds of TTT winning inside five tosses are 5/32