[joshuamorton]

You're actually being misled. Formally, you have an observed event, 10H and so you want to know the probability that your coin was fair, conditioned on having seen 10 heads. You have fair coins (F) and biased coins (B)

You have P(H) == 1/1024, P(H|B) == 1/1. And,

Then the probability that you picked a biased coin (P(B))? 1/10^1000. The probability you picked a fair coin (P(F))? (1 - 1/10^1000)

All in all, P(B|H)? Well Bayes rule tells us P(A|B) = P(B|A) * P(A) / P(B). Or in this case, P(B|H) = 1 * 1024 / 10^1000, or approximately 1 / 10 ^ 997.

> For example, if you conducted a study every week for 20 years, you'd both be extremely prolific and expect to have drawn about one wrong conclusion.

This is a prior! In the kind of experimentation I do, I run literally tens of thousands of experiments a day.