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I am not able to understand the concept of runs in terms of coin tossing. As per my question, suppose a coin is tossed x times, then we have to find the expected number of runs.

But, what is a run? Can anybody give a detailed example?

As per the question, the sequence of tosses HHHHTTHTTTHHHTHH has 7 runs. How? I am really not able to understand. I researched and found somewhere that a run is when the previous outcome is different from current, so for this case, how will it apply?

Can somebody please explain in layman terms?

Thanks

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a run is a string of h or t –  David L Feb 20 at 21:07
    
so in your case, hhhh, tt, h, ttt, hhh, t, hh are you runs –  David L Feb 20 at 21:08
1  
A run is a sequence of consecutive H or T that cannot be extended forwards or backwards. –  André Nicolas Feb 20 at 21:09
    
Why will h and t will be the runs? they are not with the same result? Should not the concept of run apply to more that one value of h/t? –  user3097157 Feb 20 at 21:09
    
HTHT has 4 runs (short ones!). –  André Nicolas Feb 20 at 21:10

2 Answers 2

up vote 8 down vote accepted

Take the full sequence and cut it at each point where there is a transition (next value is different from the previous one). Each part is a run.

 HHHHTTHTTTHHHTHH  =>  HHHH TT H TTT HHH T HH   => 7 runs

In other words, elaborating on Peter's answer, a run is a maximum-length subsequence of equal values.

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Thanks a lot, it is clear now. :) –  user3097157 Feb 20 at 21:20

A run is a sequence of equal outcomes (4 times heads , 7 times tails , etc.), so if you have n changes from head to tail or vice versa, you have n+1 runs.

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In the case of HHHHTTHTTTHHHTHH, i can understand that the following will be the cases of runs: HHHH, TT, TTT, HHH, HH. But why H and T will be also in runs? They are not sequences right? –  user3097157 Feb 20 at 21:12
    
@user3097157 they are part of the run, not a run themselves unless there is a change directly after H or T –  Rustyn Feb 20 at 21:16
    
@Rustyn, Can you please elaborate little more? –  user3097157 Feb 20 at 21:18
    
For example in this sequence: HTHHTHTHTHT. The runs are: H,T,HH,T,H,T,H,T,H,T. The runs are the chunks of tails or the chunks of heads right before there's a change. –  Rustyn Feb 20 at 21:20
    
A sequence of length 1 is still a sequence, though it is not commonly referred to as a run. –  keshlam Feb 21 at 4:31

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