QC 寫到:p(n) is the probability that you get only one successive

roll of a set of (alpha, theta and mu) at the end by n rolls.

For p(n) & F, I was considering "ONLY one successive roll of a set at the end" but not at least one.

When I was considering "at least one", I use s(n), and "S" as the generating function of s(n).

s(n)=p(0)+p(1)+p(2)+...+p(n)

S=F/(1-x)

Well, I think I mixed up the question 2 and question 3 somewhere, but it's not the main point...

QC 寫到:In my calculation of r<=1/27, I did not limit the die as fair or unfair one.

If you flip an unfair coin in order to get the pattern "(+)(-)(-)", and the chance to get (+) in one roll is p:

r=p*(1-p)*(1-p)

you will find r<=4/27.

we still don't have to consider r=1/2

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If

you flip an unfair coin/die in order to get the pattern "(+)(+)(+)",

and the chance to get (+) in one roll is p, the GF is different because

the recurrent relationship is different:

let q=1-p

p(n+4)=(1-(p(0)+P(1)+...+p(n)))*qppp

(F-pppxxx)/xxxx=pppq(1-F)/(1-x)

F=pppxxx/(1-qx-qpxx-qppxxx)

when you want "a set of 3 rolls of any kinds of conbinations only", take p=1, q=0

F=pppxxx/(1-qx-qpxx-qppxxx)

=xxx

that is, p(3)=1, p(0)=p(1)=p(2)=p(4)=p(5)=...=0

then it's so easy to see as you accept that there should be no difference for the solution for a fair or unfair situation...

how about if I flip an unfair coin to get a pattern of (+)(+)(+) with the probability of (+) is 4/5 ?? The final r = (4/5)

^{3} = 64/125 > 1/2

The main point is, r must be able to be any number between 0 and 1...

If you put r=1/3 in your previous equation, you will still get p(10)<0

See, you've changed your equation...

What is the difference to get a set of "altha,

theta, and mu" and to get a "(+),(+),(+)"?

But the new equation is still incorrect:

p(n+4)=(1-(p(0)+P(1)+...+p(n)))*qppp

how about if p = 1/2? q=1/2 too...

I assume that p(3)=1/2

^{3} =1/8 as there is no definition from your equation...

p(0)=p(1)=p(2)=0, so p(4)=p(5)=p(6)=1/16

do you think p(3)=1/2 and p(4)=p(5)=p(6)=1/16???

and also p(7)= 7/128????

Do you still think the equation is valid?