A Reward Problem to the Consecutive Heads In a Run

How many consecutive heads can we observe in a run of coin tossing of length n? Although the problem seems to be easy to answer, this would be actually a little bit tough when we try to find the solution straightforwardly. The expected number of consecutive heads in a run is (3n-2)/8 using the recursive formula.However, if we define a solitary head coin such that a head coin is isolated by neighboring tail coin(s) in a run, the problem of how many solitary heads in a run can be solved easily. The expected number of solitary heads in a run is (n+2)/8. Since the problem of solitary head coin becomes a dual problem of the above, the consequence of the problem of the consecutive heads is derived easily by considering the probability of a solitary coin appearance. Using this duality, we can solve much more complex problem such that how much the reward is expected in a run of coin tossing of length n if the reward is 2^(k-1) when k consecutive heads appears. The expected reward is (n^2+3n-2)/16. Applying this result to adaptive e-learning systems, we can design the reward to promote self-study for students.