Question

A biased coin (P(head) = p , p is not equal to 0.5) is tossed repeatedly. Find the probability that there is a run of r heads in a row before there is a run of s tails, where r and s are positive integers. Express the answer in terms of p, r and s.

Answer 1

biased coin (P(head) = p

biased coin (P(tail) = 1 - p

=  being the first hitting time of the state of r consecutive heads if we keep on tossing the coin indefinitely

=  being the first hitting time of the state of s consecutive tails if we keep on tossing the coin indefinitely

Conditioning on the first throw is a head, we have 2 possibilities:
(a) we get a further r-1 heads, in which case we stop at success; or
(b) somewhere in the next r-1 tosses we have the first tail; in which case it is as if our first toss is a tail, forgetting all those previous heads, by what is called the strong Markov property.

This gives us the equation

Conditioning on the first throw is a tail, we have 2 possibilities:
(a') we get further s-1 tails, in which case we stop at failure; or
(b') somewhere in the next s-1 tosses, we have the first head; in which case it is as if our first toss is a head, forgetting all those previous tails, by the strong Markov property.

This gives us the equation

Solving this pair of simultaneous equations gives you

Finally, remember we conditioned on the first toss