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Writing down the outcomes of a single coin flipped for t times we have a series of t outcomes, each could be T or H. A "subseries" indicates the number of the run of heads or tails in the given series. For example in the following series: $$ \text{THHHHHHTT} $$ There are three different sub series with the length of 1, 6 and 2. The question is to find the average of subseries with the length of L when flipping the coin for t times. P(t,L) is the number of times a subseries appears in t times of flipping a coin. So the question is basically to divide P(t,L) by 2^t which is the number of all possible outcomes. Note that for example in throwing for 3 times:

  • TTT 0 subs with length 1
  • TTH 1 sub with length 1
  • THT 3 subs ...
  • THH 1 sub ...
  • HTT 1 sub ...
  • HTH 3 subs ...
  • HHT 1 sub ...
  • HHH 0 subs ... Then P(3,1)= 10

I think this question might also be solvable with the "Linearity of expectations" concept but i don't know how.

Dojou
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  • Related: https://math.stackexchange.com/questions/1409372/what-is-the-expected-length-of-the-largest-run-of-heads-if-we-make-1-000-flips – Matti P. Oct 19 '20 at 11:12
  • the question in this link is different i think because it is asking for the average of max expected head runs while in this question we already know the length and the question is to find the average of the number of times the given length is expected to happen@MattiP. – Dojou Oct 19 '20 at 12:06

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