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?


  • $\begingroup$ a run is a string of h or t $\endgroup$ – David L Feb 20 '14 at 21:07
  • $\begingroup$ so in your case, hhhh, tt, h, ttt, hhh, t, hh are you runs $\endgroup$ – David L Feb 20 '14 at 21:08
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    $\begingroup$ A run is a sequence of consecutive H or T that cannot be extended forwards or backwards. $\endgroup$ – André Nicolas Feb 20 '14 at 21:09
  • $\begingroup$ 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? $\endgroup$ – Ambidextrous Feb 20 '14 at 21:09
  • $\begingroup$ HTHT has 4 runs (short ones!). $\endgroup$ – André Nicolas Feb 20 '14 at 21:10

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.


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

  • $\begingroup$ Thanks a lot, it is clear now. :) $\endgroup$ – Ambidextrous Feb 20 '14 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.

  • $\begingroup$ 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? $\endgroup$ – Ambidextrous Feb 20 '14 at 21:12
  • $\begingroup$ @user3097157 they are part of the run, not a run themselves unless there is a change directly after H or T $\endgroup$ – Rustyn Feb 20 '14 at 21:16
  • $\begingroup$ @Rustyn, Can you please elaborate little more? $\endgroup$ – Ambidextrous Feb 20 '14 at 21:18
  • $\begingroup$ 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. $\endgroup$ – Rustyn Feb 20 '14 at 21:20
  • $\begingroup$ A sequence of length 1 is still a sequence, though it is not commonly referred to as a run. $\endgroup$ – keshlam Feb 21 '14 at 4:31

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