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Homework Statement
An infinite-length binary text create from these patterns, with probability:
- "0": 0.8
- "11": 0.1
- "100": 0.05
- "101": 0.05
I temporary call a,b,c,d for P("0"), P("11"), P("100"), P("101")
Problem: create a probability table for length-3 patterns (that is, find P(x): x="000","001","010",...,"111", x is excerpted at any point of the text)
Homework Equations
The Attempt at a Solution
Some of the patterns can be determined easily: 000, 011, 100, 101, 110 with
P(000)=a3=0.512
P(011)=P(110)=a*b=0.08
However, there are no evidence about 001,010,111. So I think of context, e.g., 001 is create with 2 "0" and anything start with 1 ("11","110","101") or 2 first zero from "100" and final 1 from anything start with 1, hence:
P(001)=a2*(b+c+d) + c*(b+c+d)=0.138
With this logic, we have:
P(010)=a*d+c*d+d*(b+c+d)=0.0525
P(111)=b*(b+c+d)+(b+d)*b=0.035
However, that logic must be applied for 000,011,100,101,110 as well:
P(000)=a3+c*a=0.552
P(011)=a*b+c*b+d*(b+c+d)=0.095
P(100)=c+(b+d)*a2=0.146
P(101)=d+(b+d)*a*(b+c+d)=0.11
P(110)=b*a=0.08
And after all, the sum of (1≤x≤8),P(x)=1.2085
I know that there is something wrong with taking the context to account, but I don't really know any other ways than considering the context.
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