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So, how would you decide of the 0.02% loss at 100X odds is a reasonable bet (given variance and all that, I don't know those terms) since you have the ability to walk away at any time? Or do you only bet on things with a positive expectation? So if it were positive 0.02% would that make a difference? Isn't the variance so large it doesn't matter if there is a very slim preference for or against?

You're right. We are getting somewhere. Knowing that expected value is relevant for a single trial, doesn't tell you how relevant it is. In fact, with an expected value close to zero (or even not so close), your single trial result is dominated by the second moment (volatility or variance) and not expected value (the first moment). In the case of a single -0.02% craps bet (assuming you could do it in a single trial, which is complicated a bit by the rules of craps), we don't need things like variance or standard deviation to make a decision. Those things are only used to condense information about entire distributions, but here, the distribution doesn't need to be condensed.

Betting \$20, you either win \$20 or lose \$20 (simplifying craps again for the sake of the discussion). You have around 49.99% chance of the former and a 50.01% chance of the latter. That's it. That's all probability can tell you for a single trial. So, you are welcome to conclude that you would like to perform a single trial at those odds. It might be a "reasonable" decision for the gambler whose utility function was not risk averse. However, it would clearly be an inferior decision to the inverse single trial bet (the one made by the casino). Just because the difference in expected outcome is miniscule compared to the volatility, doesn't mean it is invalid or irrelevant. That becomes clear in the single trial case as expected value moves away from zero, and you are looking at a jar with a ratio of 341 trillion:49. In both cases, the rational choice is to pick the bet with the higher expected value, but it makes a much bigger difference in the second instance.

Sure, the casino might not win against you while still winning overall, and the volatility of your returns(luck) will dominate the expected value. The casino might also leave you with a -100% return (0 for 1), which it also won't achieve overall. Does the ability to walk away help you here? No! That's the same myth that supports the bogus Martingale progression betting systems. Why would the ability to walk away help you? It only helps you if you first buck the fractional odds against you and win. But it's not the walking away that helps you, it's the bucking the odds part! Whatever you do after that is independent of your past results. If your strategy is to continue playing until you're ahead, you are playing partial Martingale, but you do not have inifinite trials or inifinite capital, so for all times you win under this strategy, you will likely have big losses commensurate to those winnings plus some for the negative expectation.

Finally, it doesn't take all that many trips to casinos playing negative expectation games to create a 68% (1 Standard Deviation in a normal distribution) aggregate probability that your outcome will fall within a range that is completely negative. In other words, it doesn't take much for the first moment to dominate.

Let's say the gun you mention has 72 chambers. If the doomed man had a choice between two guns, on with 3 bullets in the chamber and another with 4, would he really be that better off to pick the 3 bullet gun? If I did 10 tries with each gun, is it really a good bet to guess that the 4 bullet gun is safer? In a lot of tries, yes, but not in a few. If you were to pay me cash, lots of it, to use the 4 bullet gun rather than the 3 bullet gun, assuming I had to use one or the other, I would do it for the right amount. That's the key, what is the right amount?

That's an interesting and complicated question, partly because it requires that you assign a monetary value to your life. But that isn't what we've been talking about. In my jar game, you had the choice to pick the positive or negative expectation for free! At the craps table, people pick the negative expectation for free, less entertainment and alcohol. If you are getting paid to pick the negative expectation, the expected outcome changes, and it's a wholly different question. For example, if MGM paid me \$0.50 for every \$100 I bet playing basic strategy BJ, my expectated value would be zero, no -0.5%.

Most numbers are “transcendental.” Very informally, this means that they lack a simple definition. Most numbers with names are not of this variety, so 2, Ö3, and 7/5 are not such numbers. Indeed, most people only know the names of one or two transcendental [number]s: the best known is p, the second is e. So many numbers are transcendental that if all the numbers were put in a barrel, it is good as certainty that the first number pulled out would be so. (In such a lottery always bet on the transcendental [number]s.)

Peter Borwein, Science (1994).

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