Suppose that we bet on the outcomes of coin flips in the following way. Each of us chooses a different series of three flips, and we flip a coin until three consecutive flips, in order, match your choice or mine. For example, you might pick HHT while I pick HTH, and we flip until either HHT or HTH.
Is one of us more likely to win?
My friend Shimon argues as follows. If you pick HHT and I pick HTH, a long stream of flips will sooner or later include HH, the first two flips in your choice; and at that moment you have won, because there's no way that later flips can produce my HTH streak before it produces your HHT streak. But no corresponding situation exists for the other side; should HT, the first two flips in my choice, arise, it's still possible for you to win, because the next flips could be THHT, so that the six flips HTTHHT would be another win for you.
I consider this total nonsense. It's obvious that the next three flips are as likely to be HHT, or HTT, or any of the six other possible sets of three flips. His approach of asking whether wins are possible given certain strings of past flips is mind-boggling, working without any clear sample-space and defying mathematics.
I suggested a simulation. The simulation I produced is an Excel spreadsheet that flips sets of 65 flips. In each set, it finds the first flip that begins your choice or mine, and assigns a winner. The following screen shot shows that simulation.
Blue is associated with HHT, and pink is associated with HTH. In the first set, HHT began in flips 2, 10, 16, 20, 29, 36 and 50. HTH began in flips 3, 5, 11, 13, 21 and 51. Because HHT arose first, there's a 1 in the blue vertical bar beside the first set. As you may be able to see, HHT won 8 sets of the pictured 10 sets.
In my little experiment, HHT won far more often, and often by dramatic margins.
I've looked around this stack exchange, and found a few questions almost exactly like mine – like this one and this one -- and several questions at least very similar to mine -- like this one and this one.
They all seem to believe that Shimon is right, and some sequences are more likely than others. But despite that seeming (if not very clearly stated) unanimity, and despite my own simulation, I just cannot believe it. Obviously every possible set of three flips is equally likely.
So I guess that my question is, can someone explain this at a more intuitive level, reaching the mistake that's misleading me or (I still think) Shimon?