# almost reversed 2-lag Markov chain

**R – Xi'an's Og**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**A**nother simple riddle from the Riddler: *take a binary sequence and associate to this sequence a score vector made of the numbers of consecutive ones from each position. If the sequence is ten step long and there are 3 ones located at random, what is the expected total score? *(The original story is much more complex and involves as often strange sports!)

Adding two zeroes at time 11 and 12, this is quite simple to code, e.g.

f=0*(1:10) #frequencies for(v in 1:1e6){ r=0*f#reward s=sample(1:10,3) for(t in s)r[t]=1+((t+1)%in%s)*(1+((t+2)%in%s)) f[sum(r)]=f[sum(r)]+1} f=f/1e6

and the outcome recovers the feature that the only possible scores are 1+1+1=3 (all ones separated), 1+1+2=4 (two ones contiguous), and 1+2+3=6 (all ones contiguous). With respective frequencies 56/120, 56/120, and 8/120. With 120 being the number of possible locations of the 3 ones.

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