I was led to this by the discussion on the non-random nature of prime numbers – as apparently it inspired one of the authors of the paper that noted this. I am struggling a bit with the maths of this, so hopefully writing it out might either help me grasp what is wrong with my exposition or else somebody will explain to me what I am doing wrong.
The paradox, taking H for heads and T for tails, is this – if you have a coin and toss it twice there is an equal probability (0.25) that you will see the sequence HH or HT.
But if you have two coins and toss them then, on average, it will take six tosses before you see HH but ocannly four to see HT.
I have no problem with the HT sequence – inside four tosses you can reach this via:
which a simple sum of probabilities shows comes to: , greater than half, while just three tosses would give a probability of 0.5.
So, the issue is with the HH combination. With just five tosses we can get:
Which sums to: – i.e., more than half (four tosses give us odds of 0.5).
So where is the flaw in my reasoning?