I'm not entirely sure what's being asked here. Is this asking "if we do experiment 1000001 and see k Rs in the first four trials, then what credence do you assign to the 5th trial being R?"
Or is it "if we take a random experiment out of the million and see k Rs in the first four trials, then what credence do you assign to the 5th trial being R"? This isn't the same question as the first.
Or is it something else again?
Answer: [0.111020, 0.324512, 0.5, 0.675488, 0.888980]
I will provide my solution when the market is resolved.
You do get one guarantee, though: All the experiments are Bernoulli processes. In particular, the order of the trials is irrelevant.
I think those aren't quite equivalent statements? If I pick my favorite string of bits, and shuffle it by a random permutation, then the probability of each bit being 1 is equal, the order is totally irrelevant (it was chosen at random), but it's not Bernoulli because the trials aren't independent of each other (if you know what my favorite string of bits is, you can learn the final bit as soon as you've observed all the rest.)
Correct, they are not equivalent. The second statement is a consequence of the first. I made this consequence explicit to justify my choice later on to bucket by the number of s but not their order.
The first statement, though, is also true. It's your full guarantee.
No; your distribution gives probabilities [0.253247, 0.168831, 0.155844, 0.168831, 0.253247] for the number of Rs in the first four trials. This predicts that the number of experiments with two Rs is binomially (i.e. approximately normally) distributed with mean ~155844 and standard deviation ~363, but the actual number is 161832, around 16 standard deviations away from the mean.
I have run 1,000,000 experiments. Each experiment consists of 5 trials with binary outcomes, either (for left) or (for right).
However, I'm not going to tell you how I've picked my experiments. Maybe I'm just flipping a fair coin each time. Maybe I'm using a biased coin. Or maybe I'm doing something completely different, like dropping a bouncy ball down a mountain and checking whether it hits a red rock or a white rock first--and different experiments are conducted on different mountains. I might be doing some combination of all three.
You do get one guarantee, though: All the experiments are Bernoulli processes. In particular, the order of the trials is irrelevant.
Your goal is to guess the marginal frequencies of the fifth trial. For each , you need to tell me the frequency that the fifth trial is an given that of the outcomes of the first four trials are .
For example, if every experiment is just flipping a fair coin, then the fifth trial will be an with probability , no matter what the first four are. However, if I'm using biased coins, then the frequency of will increase the more s seen.
To help you in your guessing, I have provided a csv of all the public trials. As an answer, please provide a list like [0.3, 0.4, 0.5, 0.6, 0.7] of your frequencies--the kth element of your list is the marginal frequency over the experiments with of the first four trials being .
I haven't yet looked at the frequencies myself, but I will do so shortly after posting this. If you want to test your guesses against others, I have created a market on Manifold Markets. I will resolve the market before I reveal the correct frequencies, which will happen in around two weeks, but maybe earlier or later depending on trading volume.
Good luck!