# How do I calculate the probability distribution of the percentage of a binary random variable?

I have an urn containing balls that are all either black or red. I'm interested in discovering the percentage of balls that are red. But I can only sample from the urn (without replacement), so the best I can do is calculate a probability distribution over possible percentages.

Obviously, if I've drawn no balls, I have no information, so the probability distribution is uniform from 0 to 1. But what is it once I start drawing balls?

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Theory: if $reds$ is the number of red balls you've seen and $blacks$ is the number of black balls you've seen, then the distribution is:
$Beta(reds+1, blacks+1)$
This starts out as $Beta(1,1)$ which is the uniform distribution we want. As we see blacks, it shifts toward zero; as we see reds, it shifts toward one.