HERC777 wrote: > there's about 1,000,000 combinations of a sequence of 20 coin flips > 2^10 = 1024 > 2^20 = a million > > so on average they should copy your sequence up to the 1st 20 flips.. > > Agree?
No. Jesus. First of all, you have to know what you are averaging OVER. You are overaging over FLIPPERS, or flip-sequences. You are averaging over a million things. For each of these million strings (of H or T), you want to know how long its prefix-matching-you is. The number of flippers matching-you-so-far starts out at a million after 0 flip-positions and falls by half with every flip(in time) thereafter. Half the million will flip opposite from you on the first flip and so will have a prefix-matching-you of length 0. 1/4 of them will match you on the first flip and miss on the 2nd, and have a match-you length of 1. 1/8 of them will match you on the first two but miss on the 3rd, and have a match-you length of 2. On the 20th flip, the EXPECTED number of flippers continuing to match you all this way will be less than 1. Furthermore, the AVERAGE, over all flippers, of the initial-length-of-match-you-prefix, is ALSO LESS THAN 1.