In a series of Bernoulli trials with success probability p, find the probability that k successes in a row occur prior to n failures in a row.

(Hint: Let  be the event of interest, namely,  ∶ k successes in a row prior to n failures in a row. Consider now the (conditional) probabilities p1 = P(|the first trial is a success), p2 = P(|the first trial is a failure).

If we know that the first trial resulted in a success, for  to occur we must have that

• the trials numbered 2, 3,…, k all result in a success, or

• there is a first failure at the mth trial for some 2 ≤ m ≤ k, and then  occurs starting from a failure.

This gives us one equation relating p1 with p2. We can get another one in a similar way. Thus, we obtain both p1 and p2 and hence the required probability.