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)