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As I think you understand by now, I don't see these as similar betting oportunities, but I believe you do. Right?

No! I completely agree that the number of trials makes a significant difference in the overall characteristics of the bet. But where we disagree, I think, is that I think they differ because of differing variances, and not differing expected outcomes! If you popped down to Atlanta and bet me a beer that you'd pick one of the 49 red in my 51G/49R jar, I would take the bet, because my expected outcome is positive, and because the massive variance in that trial is diversified by the rest of my "invested" capital (assuming I have more than $4.50 to my name). In finance academic lingo, the bet would have high idiosyncratic risk but a low market risk (actually, market risk would be zero, since there's no correlation between you picking marbles and the rest of my capital, unless you happen to be my boss).

But if you bet me everything I had, I wouldn't make the bet, because despite the positive expectation, that massive single trial variance (risk) would overtake the utility of the favorable odds (for me). Now, if you came down and wanted to bet everything I had on a single trial using my jar with 341 trillion green and 49 red, I would do it in a heartbeat, because my expected outcome is so high as to make even the high single trial variance favorable (to me).

The bet I would never make (unless I considered it paying for entertainment), however, is on picking one of the 49 red chips. Not for 1 trial or a billion trials. That's why I've never bought a lottery ticket.

If expected value did not apply to single trials, what advice would give this prisoner who is offered two guns by his captors with which he's got to play Russian Roulette. In the first gun 11 of 12 bullets fill the chamber. In the second, only 8 of 12 bullets are loaded. He only has to fire once, and he can go free if he survives. (Assume there's no trickery by the captors.) Pick a gun. Is expected outcome irrelevant for this single trial?

Sorry. I said I'd shut up. I will. I think. It's a sickness.
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