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June 29th, 2010, 06:20 PM   #1
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toss coin until the first run of r consecutive tails occurs

if a single
coin, with probability q = 1 - p of obtaining tails on each toss, is tossed repeatedly
until the first run of r consecutive tails occurs, How is the expected number of tosses????.....and how is the Probability tha we
toss the coin n-times(n>=r) ?????????? THANK YOU VERY MUCH.
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June 30th, 2010, 12:09 AM   #2
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Re: toss coin until the first run of r consecutive tails occurs

Hi, nice to meet you. I'm new to this stuff.

I think I can't help you much, I could only find the following :

let A be the event : thrown r times tail consecutively,

prob. of A knowing n tosses were thrown were :

p(A|n)={
0, if n<r
q^r, if n=r
q^rp, if r<n<2r+1
q^rp^2, if 2r+1<=n<3r+1
...
then using Bayes thm we could maybe find : p(n|A) which is : probability of n tosses knowing that r consecutively happened.

bye.
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June 30th, 2010, 06:56 AM   #3
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Re: toss coin until the first run of r consecutive tails occurs

After the first r-1 tosses, you have roughly a q^r chance per flip of getting r tails in a row. There was a (long) thread about this on the Mersenne forums if you need more detail.
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June 30th, 2010, 06:47 PM   #4
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Re: toss coin until the first run of r consecutive tails occurs

I tried solving this problem using Markov chains. Consider the first r throws as leading to one of the r+1 initial states {"...H", "...HT", "...HTT", "...HTTT", "...HTTTT", ... "T...TTTT"} and create a substochastic matrix with r rows:



Then the expected number of throws is
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June 30th, 2010, 08:38 PM   #5
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Re: toss coin until the first run of r consecutive tails occurs

If p=q=, the answer is .
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July 1st, 2010, 06:54 AM   #6
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Re: toss coin until the first run of r consecutive tails occurs

Hi,

do you know how to link this result to the probabilities in post 2 ?

this gave : <n>=r+p(r+1)+p^2(2r+1)+..p^k(kr+1) but I couldn't achieve to sum this up.

(r tail+ 1head + (any r-1)+ 1head+...i thought maybe the develop. of (1-t)^{-1}=1+t+t^2+t^3, could help ? )

thanks.

Bye.
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July 1st, 2010, 11:49 PM   #7
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Re: toss coin until the first run of r consecutive tails occurs

>Erratum :

sorry : r+p(r+1+r+2+...r+r)+...p^m(mr+mr+1+...mr+r)+...= ?
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