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Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are independent.

The maximum number of trials is $n$, but if there are $k<n$ consecutive successes the experiment ends.

What is the probability distribution?

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Let $a_n$ be the required probability.

Then $a_{n+1} = a_n+(1-a_{n-k})p^k(1-p)$

Where $a_0=a_1=a_2=...=a_{k-1}=0,a_k=p^k$

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