Proposition: A chance pattern is possible

For an independent Bernoulli correctness process with success probability $q$, one five-trial window satisfies the criterion with probability

At $q=1/4$, this is $1/64$. At $q=3/4$, it is $81/128$.

For $H\ge 5$, the probability of at least one qualifying window is at most

Proof. Sum the probabilities of exactly four and five successes. Use a union bound over $H-4$ windows. Their overlap does not affect the validity of that upper bound.