One more -- Coin Flipping

Lets say I had a coin that wasn't fair. I flipped the thing 1000 times and I got 600 heads. How "confident" would I be that the "true" probability of getting heads was 55% or more? How confident am I that the "true" probability of getting heads is 60% or more?

If I kept flipping the coin and heads kept coming up 60% of time, how many times would I have to flip it so that I am 95% sure that the "true" probability of heads is 60%? If I flipped the coin a million times and got 600,000 heads, then I'd be pretty sure.

Let me try to simplify this. How many times would I need to flip a coin, and what would I have to see, to conclude, to 95% certainty, that heads has a 55% chance of coming up.

This is an interesting question and actually a lot more complex than you probably think.

The answer is that it depends on your initial estimate of the probability that the coin is fair, the coin is 55%, etc.

If you initially decide the chances are 100% that the coin is fair, a million trials would not be enough to convince otherwise. If you initially decide that the chances are 1 in a million that the coin is biased towards 55% heads, and otherwise fair, it turns out that after your 1000 trials the chances are about 22.7% that the coin is fair.

If you estimate 1 in a 1000 chance the coin is biased towards 55% heads, the chances are well less than 1% after the 1000 trials that the coin is fair.