- #1
Jamin2112
- 973
- 12
My sequence is a function f:Z+→{H,T} to represent heads and tails. It is defined by
f(1),f(2),f(3)f(4),f(5)f(6), ...
= H,T,HH,HT,TH,TT,HHH,THH,HTH,...,
where you can see that I'm just taking all heads-tails sequences of length 1, then of length 2, etcetera. This ensures
(1) that the long term frequency of heads is 1/2.
(2) that every finite heads-tails sequence is contained the infinite sequence thus defined.
The proofs of (1) and (2) are trivial. There's one more property that I'm not sure I have in my sequence, so I'd like help proving it (if it's true ...). I need that every heads-tails sequence of length N occurs with equal long-term frequency. So, for example, I should see HT just as often as TH when I look left to right at
HTHHHTTHTTHHHTHHHTH...
Thoughts?
f(1),f(2),f(3)f(4),f(5)f(6), ...
= H,T,HH,HT,TH,TT,HHH,THH,HTH,...,
where you can see that I'm just taking all heads-tails sequences of length 1, then of length 2, etcetera. This ensures
(1) that the long term frequency of heads is 1/2.
(2) that every finite heads-tails sequence is contained the infinite sequence thus defined.
The proofs of (1) and (2) are trivial. There's one more property that I'm not sure I have in my sequence, so I'd like help proving it (if it's true ...). I need that every heads-tails sequence of length N occurs with equal long-term frequency. So, for example, I should see HT just as often as TH when I look left to right at
HTHHHTTHTTHHHTHHHTH...
Thoughts?