I'm replying to myself. Warning: serious math ahead.

This is part of a class of statistical problems that I really want to figure out how to solve, because I feel like understanding problems like this will give me a much, much stronger foundational understanding of statistics. For example, can anyone here explain how sampling a normally-distributed population with unknown mean and unknown variance brings forth the Student-T distribution? No explanation that I have read makes any sense on a pre-computation, intuitive level of understanding.

So let me at least set up the foundation for the math needed to answer the question: "What are the odds that Netherlanders are actually happier than Americans regarding their jobs, given that a sample of 1000 random Americans produced 93% whom are happy with their jobs, and a sample of 650 Netherlanders produced 91% whom are happy with their jobs".

The setup: let's imagine a multiverse. Inside this multiverse are an infinite number of universes, and inside each universe, x% of Americans are happy with their jobs. The value of "x" varies between each universe; for each universe, there is a single, unique value of "x", and for each value of "x", there is a single, unique universe.

Let's imagine that when we see the results of a survey of 1000 Americans, we do not know which universe it came from. It's possible that exactly 930 out of 350,000,000 Americans are happy with their job (essentially 0%), but that random selection of 1000 Americans just so happened to pick those 930. It's possible that 99% of Americans are happy with their jobs, but the sample just so happened to randomly pick up too many unhappy workers.

If we have a universe where 50% of the American population is happy, then there is a 1000-Choose-930 * 0.50^930 * 0.50^70 chance that a random sample of 1000 will have 93% of the sample will be happy with their jobs (binomial distribution; the odds that 10 coin flips will land on 5 heads depends partially on the number of ways 10 coins can be deliberately arranged so that 5 specific coins will be heads up). 1000-Choose-930 * 0.50^930 * 0.50^70 = 6.5710^-193. If we have a universe where 93% of the American population is happy with their jobs, then there is a 1000-Choose-930 0.93^930 * 0.07^70 chance that a random sample of 1000 will have 93% of that sample be happy with their jobs. 1000-Choose-930 * 0.93^930 * 0.07^70 = 0.0494. In general, when we have a universe with x% of the population happy with their jobs, the odds that a random sample will show 93% happiness is 1000-Choose-930 * x^930 * (1-x)^70.

We now need to step back into our multiverse and realize that we're sampling 1000 people from each universe with equal probability, and then collecting the number of "samples of 1000 people show 93% happiness" instances into a single pot. We need to ask ourselves, "given that we have a sample of 1000 people showing 93% happiness, what are the odds that we are in a universe where x% of the American population is happy?". The answer to that is: "The number of times a universe with x% happiness produces a sample of 1000 with 93% happiness, divided by the number of times all the universes in the multiverse produces a sample of 1000 with 93% happiness.

The percentage of "93% of sample of 1000 are happy with their jobs" samples in a universe with x% happiness, compared to all the other equally-likely-to-be-chosen universes, in a situation where each universe has been sampled 10^99999 times, is approximately: (1000-Choose-930 * x^930 * (1-x)^70 * 10^99999) / sum(i=0;i<100;i+=1 { 1000-Choose-930 * i^930 * (1-i)^70 * 10^99999 } ), assuming that there are only 100 universes in a multiverse where "x" takes on only integer values. For a bit more rigor, we have to allow x to take on all real values, and recognize that the probability of picking a single value of "x" out of a continuum of real values from 0 to 100 is effectively 0%; what we ultimately have to do is calculate the probability that the universe we inhabit rests within a range of values of x.

So, bringing a bit of calculus into the fold: The percentage of "93% of sample of 1000 are happy with their jobs" samples in a universe with x% happiness (where x is between numbers x1 and x2), compared to all the other infinite equally-likely-to-be-chosen universes between x=0 and x=100, is: integral(x1 through x2: 1000-Choose-930 * x^930 * (1-x)^70) / integral(0 through 100: 1000-Choose-930 * i^930 * (1-i)^70)

Likewise, if we create another multiverse of universes where y% of Netherlanders (with varying values of y) are happy about their jobs, then the odds that a random sample of 650 Netherlanders will have 91% of that sample confess happiness with their job is: 650-Choose-592 * y^592 * (1-y)^58. So if we consider all universes equally likely, and we are given a sample of 650 people, 91% of whom are happy with their jobs, then the odds that we are living in a universe where the real, actual percentage of Netherlanders are happy with their jobs lies in between values y1 and y2 can be calculated. That calculation is: integral(y1 through y2: 650-Choose-592 * y^592 * (1-y)^58) / integral(0 through 100: 650-Choose-592 * y^592 * (1-y)^58).

My original question was this: what are the odds that the original survey was a statistical fluke, and that Netherlanders are happier than Americans with their jobs? Well, if 0% of Americans are happy with their jobs, then we need to know the odds of 0%-100% of Netherlanders are happy with their jobs. If 1% of Americans are happy with their jobs, then we needs to know the odds of 1%-100% of Netherlanders are happy with their jobs. If there is a 50% chance that 0% of Americans are happy with their jobs, and also a 50% chance that 1% of Americans are happy with their jobs, then the odds that Netherlanders are happier with their jobs than Americans is 50% times the odds of 1%-100% Netherlander satisfaction, plus 50% times the odds of 2%-100% satisfaction. Generalizing this pattern, we get this equation: The odds of Netherlanders being happier = The odds of x% of Americans being happy with their jobs * the odds of x%-100% of Netherlanders being happy with their jobs.

The odds that x% of Americans are happy with their jobs is: integral(x through x+epsilon: 1000-Choose-930 * x^930 * (1-x)^70) / integral(0 through 100: 1000-Choose-930 * i^930 * (1-i)^70). The odds of x%-100% of Netherlanders are happy with their jobs is: integral(x through 100: 650-Choose-592 * y^592 * (1-y)^58) / integral(0 through 100: 650-Choose-592 * y^592 * (1-y)^58). Now we need to multiply those values together and take the limit as epsilon goes to zero:

lim (as epsilon -> 0): integral(0 through 100: {integral(x through x+epsilon: 1000-Choose-930 * x^930 * (1-x)^70) / integral(0 through 100: 1000-Choose-930 * i^930 * (1-i)^70)} * {integral(x through 100: 650-Choose-592 * y^592 * (1-y)^58) / integral(0 through 100: 650-Choose-592 * y^592 * (1-y)^58)}.

Someone please evaluate that expression. Or at least check my math/logic.