Just how many coin tosses would it take?

I recently came in pocession of a fereign currency coin which seemed to be biased(to heads say). On noticing that the coin seemed asymmetric, I set up the experiment and started tossing the coin. Here is how things went :

after 2 coin tosses: 2H, 0T : Prob(H) =1

after 30 coin tosses: 15H, 15T : Prob(H) = .5

after 3000 coin tosses : 1492H, 1508T : Prob(H) = .497333333333333333333333333333

Now to state the obvious, the first result was a joke. In the third case, notice how I've conveniently written the number to more than 10 places of decimal. At some point, I had to stop and think, does it make any sense to claim the number to 10 places of decimal with just 3000 observations?

One thing I could agree upon was that, higher the number of observations, higher is the precision, but how much?

I guess I want to quantify precision as a function of number of observations.

Please share your thoughts on the problem, any suggestion would be appreciated.

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$