I have several suggestions:
- Do not try the compression-based attempt listed above, it isn't likely to work
- You can't tell from a single test if a stretch of 500 heads followed by 500 tails is non-random: you'd have to permute or at the very least replicate
- You can (should!) test higher-order statistics. Right now, you perform a chi-squared distributional test on 0-th order statistics (number of heads, number of tails). Instead, perform that test on the number of pairs (e.g. 1st order statistics): HH, HT, TH, TT. You can extend that to higher-orders if you are really interested, but first order should be enough to find bias, and easy to implement.
So in summary, I recommend you perform multiple (100s) tests on 0th, 1st, and 2nd order statistics.
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