[jules]

The probability of heads is

    0.999*0.5 + 0.001*1 = 0.5005

The important thing is that as you observe heads from the coin, you learn something about the coin. As you observe more heads it is less likely to be a fair coin and more likely to be the coin with double heads. This doesn't change anything about the coin, but it changes something about what you know about the coin. See here for the correct answer: https://news.ycombinator.com/item?id=7000523

[joezydeco]

Since the question simply asked what p(heads) was on the next flip, it seems our answers match. Thanks!

[jules]

Our answers to your question may match, but I very much doubt that our answers on the original question match. The crux is that your question is not equivalent to the original question. If you are interested in learning why that is I can explain it further, but it doesn't look like you are?

[joezydeco]

Okay yeah, I'm stupid for pressing on while crntaylor already gave his answer. I'll shut up now.

[baddox]

Not at all. The answer is about 0.75, and stated in the answer jules linked to.