Re: is the EV for this game really infinity?
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If you are just going to play the game one time, you should expect the game to last two rounds and expect to win $4.
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I think your confusion may be because are using a standard English definition of the word 'expect,' and not the precise probability theory definition of the word 'expectation.'
To illustrate: If a friend of mine offered to roll a single dice and pay me $1 if it came up 1-5, but $100 if it came up 6, I would "expect" to win $1, but my "expectation" would be $17.50
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