Comment by ska

This gets into interesting stuff!

So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful.

> Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads.

This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6%

A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw".

It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen).

Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.