Calculating probability for a never ending situation which is recurring.
Problem: A coin tossing game between two friends (of course Alice & Bob may be Eve is also there watching them play).
Find probability of Alice winning if both Alice and Bob has two weighted coins which has
probability of coming heads.
Assume Alice plays first and whoever gets Head first wins.
P(A) = Probability of Alice winning the game = Probability of Alice coin to show head on first throw + (Probability of Alice count to show Tail on first throw)(Probability of Bob coin to show tail)*Probability of Alice count to come head + .. …
H + TT H + TT TT H
P(A) = a + (1-a)(1-b) P(A)
=> P(A) = a / (a(1-b) + b)