If the blog author is reading, some notes for improvement: - Your odds calculation is likely wrong. You assumed from the word "odds" that "odds ratio" was mean

If the blog author is reading, some notes for improvement: - Your odds calculation is likely wrong. You assumed from the word "odds" that "odds ratio" was meant, (Odds=3 meaning "odds 3:1 against" corresponding to p=25%) but the phrase is "approximate odds 1 in X" (Odds=3 meaning "odds of 1 in 3 to win" meaning 33%) and recalculating results in the remarkably exact expected value of $80 which seems intentional? - You phrase things in terms of variances, people will think more in terms of standard deviations. So 3.5 ± 1.7 vs $80 ± $12,526. - Note that you try to make a direct comparison between those two but the two are in fact incomparable. The most direct comparison might be to subtract 1 from the die roll and multiply by $32, so that you have a 1/6 chance of winning $0, 1/6 of winning $32, ... 1/6 of winning $160. So then we have $80 ± $55 vs $80 ± $12,526. Then instead of saying you'd need 50 million more lottery tickets you'd actually say you need about 50 thousand more. This is closer to the "right ballpark" where you can tell that the whole lottery is expected to sell about 10,200,000 tickets on a good day. - But where an article like this should really go is, "what are you using the numbers for?". In the case of the Texas lottery this is actually a strong constraint, they have to make sure that they make a "profit" (like, it's not a real profit, it probably goes to schools or something) on most lotteries, so you're actually trying to ensure that 5 sigma or so is less than the bias. So you've got a competition between $20 · n and 5 · $12,526 · √(n), or √(n) = 12526/4, n = 9.8 million. So that's what the Texas Lottery is targeting, right? So then we would calculate that the equivalent number of people that should play in the "roll a die linear lottery" we've constructed is 187, call it an even 200, if 200 people pay $100 for a lottery ticket on the linear lottery then we can pretty much always pay out even on a really bad day. - So the 50,000x number that is actually correct is basically just saying that we can run a much smaller lottery, 50,000 times smaller, with that payoff structure. And there's something nice about phrasing it this way. - To really get "law of large numbers" we should actually probably be looking at how much these distributions deviate from Gaussian, rather than complaining that the Gaussian is too wide? You can account for a wide Gaussian in a number of ways. But probably we want to take the cube root of the 3d cumulant, for example, try to argue when it "vanishes"? Except given the symmetry the 3rd cumulant for the die is probably 0 so you might need to go out to the 4th cumulant for the die -- and this might give a better explanation for the die converging more rapidly in "shape" to the mean, it doesn't just come close faster, it also becomes a Gaussian significantly faster because the payoff structure is symmetric about the mean.