Define “E j ” as the event that there is a run of K successes (heads) in the first j trials. Define “A j ” as the event that the last K terms of {X i } j are T followed by K-1 H’s ({X i } j = X 1 X 2 X 3 X 4 …THHH…HHH). If you have a run of K heads in the first j trials, of course you’ll have a run in the first j+1 trials. When E≥1, that means that it’s pretty likely that there’s at least one run of K heads. If this does not happen, then at least one tail must occur within the first K coin flips.

When you want to outsmart the world, you turn to the facts. And the facts are in the science.