How the Beta Distribution Connects Sorting and Tossing Coins

A question inspired by an interview with a San Francisco-based startup

Say you have a coin and know nothing about the probability of heads. It’s reasonable to assume it’s distributed uniform between 0 and 1 (no reason to prefer any probability over any other since we haven’t seen any data). Now, you toss it 10 times and see 6 heads and 4 tails (while 60% is the most logical estimate, it could also be a fair coin). How does the distribution of the probability change in light of this…