daveylibra
New member
- Joined
- Nov 9, 2017
- Messages
- 16
Consider a random sequence, length n, of 0s and 1s, eg ( 0,0,1,1,0,1,1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,..... )
Let us denote a sequence of one 0 or 1, as S1. A sequence of two 0s or 1s as S2 etc.
I wish to prove that S1 > (S2+S3+S4+...)
ie, the total number of S1 will probably be more than the total number of all other sequences, S1+S2+S3 etc.
Further, each sequence starts after a "switch": ...0,1 or ...1,0. If we bet after a "switch" for an S1 to form, ie ...0,1,0 or ...1,0,1
prove that we have an advantage over randomly betting.
In trying to solve this, I read that S1 = n/4 + 1/2 and that S1+S2+S3.... = n/4 but I am struggling to prove this.
I have also wrote a computer simulation that backs this up for large n, but cannot find mathematical proof, so any help appreciated!
Let us denote a sequence of one 0 or 1, as S1. A sequence of two 0s or 1s as S2 etc.
I wish to prove that S1 > (S2+S3+S4+...)
ie, the total number of S1 will probably be more than the total number of all other sequences, S1+S2+S3 etc.
Further, each sequence starts after a "switch": ...0,1 or ...1,0. If we bet after a "switch" for an S1 to form, ie ...0,1,0 or ...1,0,1
prove that we have an advantage over randomly betting.
In trying to solve this, I read that S1 = n/4 + 1/2 and that S1+S2+S3.... = n/4 but I am struggling to prove this.
I have also wrote a computer simulation that backs this up for large n, but cannot find mathematical proof, so any help appreciated!