So, let’s look at the posterior (outcome) probability density function after the next two tosses. First a head:
Now we can see the results near the extreme possibilities (ie H= 1, H = 0) being eliminated completely. Interestingly, although, in reality, the evidence (two heads, two tails) points towards a fair coin, ie H = 0.5, the pdf here peaks with H below 0.5, reflecting our earlier ‘estimate’ of a coin biased towards tails.
Now, another tail:
Now a head:
Here the peak is becoming pronounced (and it also looks like the splines used to create the graph are showing signs of Runge’s phenomenon).
And the final, tenth, toss:
This does suggest a bias but the pdf is still quite broad – so there is lots of room for further correction. Indeed the thing that strikes me most is how little the suggested bias is after five heads in a row.
- Is my dime fair? (cartesianproduct.wordpress.com)
- Taylor expansion of a probability density function (cartesianproduct.wordpress.com)
- How to turn a biased coin into a Fair Coin (carlos.bueno.org)
- How to create a biased coin and prove it with math (izbicki.me)
- The binomial distribution, part 2 (cartesianproduct.wordpress.com)
- Burn the mathematics (worthwhile.typepad.com)