本文记录了100万次独立实验的结果，每次实验包含5次二元结果的试验（L或R）。实验的生成方式未知，可能涉及公平或有偏的硬币抛掷，甚至更复杂的随机过程。所有实验均遵循伯努利过程，试验顺序无关紧要。文章旨在预测在已知前四次试验中出现k次R（k=0, 1, 2, 3, 4）的情况下，第五次试验出现R的边际频率。作者提供了一个包含所有公开试验数据的CSV文件，并鼓励读者通过Manifold Markets进行预测。

📊 **实验设计与不确定性**：作者进行了100万次包含5次二元试验（L或R）的实验，但故意隐藏了实验生成机制，暗示可能存在多种复杂的随机过程，增加了预测的挑战性。

🎲 **伯努利过程与顺序无关性**：所有实验均遵循伯努利过程，这意味着每次试验的结果是独立的，且试验的顺序不会影响整体的概率分布，为预测提供了理论基础。

📈 **预测目标：第五次试验的条件频率**：核心任务是根据前四次试验中R出现的次数（k=0至4），预测第五次试验出现R的边际频率，即P(第五次为R | 前四次有k次为R)。

📄 **数据支持与社区互动**：作者提供了包含所有公开试验数据的CSV文件，供分析参考，并设立了Manifold Markets上的预测市场，鼓励参与者交流和测试各自的预测能力。

Published on October 12, 2025 10:09 PM GMT

I have run 1,000,000 experiments. Each experiment consists of 5 trials with binary outcomes, either $L$ (for left) or $R$ (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 $k = 0, 1, \dots, 4$, you need to tell me the frequency that the fifth trial is an $R$ given that $k$ of the outcomes of the first four trials are $R$.

For example, if every experiment is just flipping a fair coin, then the fifth trial will be an $R$ with probability $1/2$, no matter what the first four are. However, if I'm using biased coins, then the frequency of $R$ will increase the more $R$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 $k$ of the first four trials being $R$.

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!

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