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.