There’s actually some really cool math developed about situations like this one. Large deviation theory describes how occurrences like the 1,000,004 red / 1,000,000 blues one become unlikely at an exponential rate and how, conditioning on them occurring, information about the manner in which they occurred can be deduced. It’s a sort of trivial conclusion in this case, but if we accept a principle of maximum entropy, we can be dead certain that any of the 2,000,004 red or blue draws looks marginally like a Bernoulli with 1,000,004:1,000,000 odds. That’s just the likeliest way (outside of our setup being mistaken) of observing our extremely unlikely outcome.
There’s actually some really cool math developed about situations like this one. Large deviation theory describes how occurrences like the 1,000,004 red / 1,000,000 blues one become unlikely at an exponential rate and how, conditioning on them occurring, information about the manner in which they occurred can be deduced. It’s a sort of trivial conclusion in this case, but if we accept a principle of maximum entropy, we can be dead certain that any of the 2,000,004 red or blue draws looks marginally like a Bernoulli with 1,000,004:1,000,000 odds. That’s just the likeliest way (outside of our setup being mistaken) of observing our extremely unlikely outcome.