distribution flip a coin 1000 times

Solution 1:

If the coin were fair, then the standard deviation for $1000$ flips is ${1\over2}\sqrt{1000}\approx16$, so a result with $600$ heads is roughly $6$ standard deviations from the mean. If you're familiar with Six Sigma, you'll have grounds for suspecting the coin is not fair.

Solution 2:

You can bound the $p$ value as $$ \sum_{i=600}^{1000}\binom {1000}{i}2^{-1000}\le401\binom {1000}{600}2^{-1000}=1.9\times10^{-8} $$ which is very small. So I wouldn't say it is fair. There are more precise ways but since $600$ so far off the expected value, this rough method is sufficient.

Solution 3:

Just to add to Barry's Cipra answer: Your question follows The Binomial Distribution, hence:

$\mu=np={1\over2}*1000=500 $

and $\sigma=\sqrt{np*(1-p)}=\sqrt{1000*0.5*(1-0.5)}=15.8$

600 heads means you're looking at over 6 sigma! So to put it in perspective, with +3 sigma you're in the 99.7th percentile. Conclusion: coin is unfair.