String Probabilities

How do you determine if a pattern found in a text is significant?
One method is to determine the probability of the pattern occuring if the text is random.
However, these probabilities are not always obvious.
Consider the probability of the patterns AA and AT in (uniform i.i.d.) strings over the alphabet A,T of various length:
- In strings of length 2, both AA and AT occur with probability 1/4:
  
  AA AT
  AA AA
  TA TA
  TT TT
  AT AT
- In strings of length 3, however, AA occurs with probability 3/8 and AT occurs with probability 4/8:
  
  AA AT
  AAA TAA AAA TAA
  ATA TTA ATA TTA
  ATT TTT ATT TTT
  AAT TAT AAT TAT
- In texts of length 4, however, AA occurs with probability 8/16 and AT occurs with probability 11/16.
  
  AA AT
  AAAA ATAA TAAA TTAA AAAA ATAA TAAA TTAA
  AATA ATTA TATA TTTA AATA ATTA TATA TTTA
  AATT ATTT TATT TTTT AATT ATTT TATT TTTT
  AAAT ATAT TAAT TTAT AAAT ATAT TAAT TTAT