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I had a bit of a thought experiment today to which I don't know that answer to myself so I hoped someone here could enlighten me.

Let's say we toss an idealized coin, the result being either $0$ or $1$ with equal probability. There are exactly 100 tosses and the amount bet is always the same, but the player can decide to quit whenever they want. Max started the game. We are currently at toss 51. Max won 30 tosses and lost 20. Should Max continue?

The reason I have no idea how to solve that is the following. Technically the chance is always 50% for each toss. But over time the chance will also always adjust to 50% and Max is currently at 66%. Therefore he is currently in the plus and the likelihood to make further winnings is lower than the likelihood of going back to his original starting capital.

So could anyone explain and what implications this has for gambling in general?

Rodrigo de Azevedo
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uncanny
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  • Your fourth paragraph is not correct. The previous trials have no impact on the likelihood of the results of future trials. – Robert Shore Feb 02 '21 at 09:57
  • Could you explain that with proof in an answer please? In this case we are just recording a state in the overall section of exactly 100 trials. – uncanny Feb 02 '21 at 10:16
  • @RodrigodeAzevedo The problem states that we're using an idealized coin "with the chance of being either [$0$ or $1$] always exactly $50$%." That implies independent trials. – Robert Shore Feb 05 '21 at 08:02

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