Good question, and one that sometimes comes up when I ask it in interviews. You are tossing the same coin 10 times.
A the beginning, there's a 1/1000 chance that you pick a double-headed, and a 999/1000 chance you pick a fair coin.
A fair coin would act the way you've observed 1/1024 times. A double-headed coin would act that way 100% of the time.
(This is where I get fuzzy): Given what you've observed, there is a (1000+1024)/1024 = 0.506 chance that the coin is double-headed. There is a 0.494 chance that it's fair.
A double-headed coin would come up heads next 100% of the time. A fair coin would come up heads 50% of the time. So, 0.506 x 1 + 0.494 x 0.5 = 0.753.
How far off am I?
Anyway--assume I think all that aloud in an interview. What does that tell you about the candidate?
I would give points for just asking that question, because many people bound by conventional thinking wouldn't dare to ask it, accepting default assumption that you can't. I'm not saying this says anything about your ability to solve the problem, but asking the question is a good sign of a supple mind.
There's a .999 chance you have a fair coin and a .001 chance you have the rigged coin.
(0.999 * 0.5) + (0.001 * 1) = 0.5005.
Seems too simple, but a coin is a coin, right?
So figure the p(heads) for the coin and ignore the previous history. Overthinking it is why this makes a good FizzBuzz problem.
Suppose that the jar contains 500 double-head coins and 500 double-tail coins. You pull a coin from the jar, flip it 10 times, and get 10 heads. What is the probability it will come up heads next time?
Let's also say that every time you flip the coin and it comes up tails, you win $5. And every time you flip the coin and it comes up heads, you lose $1.
Clearly, this would be a great game to have the opportunity to play, if the coin is fair. Every time you flip you either win $5 or lose $1, so your profit, on average, is $4 per flip.
You've flipped it 10 times so far, and it's come up heads every time, and you've lost $10.
After you're $10, $100, $1000, or $10e100 in the red, without ever seeing a win, when do you change your mind about playing this game?