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Question about the Martingale system?

I'm trying to show my father that if you were to use the Martingale system on a COIN FLIP, on average, than would the same amount of money (for example after a million times). The Martingale system is to bet $ 1 on the first lap then if you lose you double bet, then you lose that double bet again. In the end this translates into small wins, but loses a lot when you finally lose. My logic is that if there was a $ 1 bet starting and the boundary of the house was $ 4, there are 8 possibilities of coin tosses: HHH HHT HTH HTT THH THT TTH TTT A of the possibilities is the loss of Sytem and 7 of them involving a victory. So therefore, 7 out of 8 times you win $ 1, and 1 in 8 times loses $ 7. Ergo, on average, you come out with even money. How is my theory wrong? By the way, I know that does not work in a casino for the space 0 at roulette so for Please do not tell me I should not play this way. I do not play at all.

While stops after winning (otherwise, we would still have to talk betting strategy in this case), then I think that is correct. Another way to see this: At every turn, won half of the possible pitches. every victory you win each loss and lost. In general, this type of phenomenon will occur whenever you have a fair game to pay exactly the bet.