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Consider an (unfair) coin with probability of heads P(H) = p.
Consider the events:
A = "(r+s) coin tosses result in r or more heads"
B = "tossing the coin repeatedly, until a total of r heads appear, results in a total s or fewer tails"

It holds that
P(A) = P(B).

Or in "math":
If X~Bin(s+r, p) and Y~NBin(r, p) then P(X ≥ r) = P(Y ≤ s)

Proof:
P(A) = P(H appears ≥ r times in s+r tosses)
= P(H appears r times in ≤ s+r tosses)
= P(T appears ≤ s times before H appears r times)
=P(B)

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