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June 29th, 2010, 06:20 PM  #1 
Member Joined: Jun 2010 Posts: 64 Thanks: 0  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 ntimes(n>=r) ?????????? THANK YOU VERY MUCH. 
June 30th, 2010, 12:09 AM  #2 
Member Joined: Jun 2010 Posts: 80 Thanks: 0  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(An)={ 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(nA) which is : probability of n tosses knowing that r consecutively happened. bye. 
June 30th, 2010, 06:56 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: toss coin until the first run of r consecutive tails occurs
After the first r1 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.

June 30th, 2010, 06:47 PM  #4 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  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 
June 30th, 2010, 08:38 PM  #5 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: toss coin until the first run of r consecutive tails occurs
If p=q=½, the answer is .

July 1st, 2010, 06:54 AM  #6 
Member Joined: Jun 2010 Posts: 80 Thanks: 0  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 r1)+ 1head+...i thought maybe the develop. of (1t)^{1}=1+t+t^2+t^3, could help ? ) thanks. Bye. 
July 1st, 2010, 11:49 PM  #7 
Member Joined: Jun 2010 Posts: 80 Thanks: 0  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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