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A person is asked to perform a task for a number of times and each time the researcher judges whether the performance is a success or a failure. The person continues performing the task until he/she has been succesful for five consecutive times, in which case the person in said to have reached the target criterion. For each person, the number of trials foregoing to the series of five successes in a row is registered and this number is the value of the dependent variable Y for that person. E.g. a person could have the following sequence of successes (1) and failures (0):

01000111011110011111

This person's Y value is 15, because the last value 0, right before the final sequence of five successes "11111" in a row, is at position 15 from the left. For the person below, with sequence

10111011111

the Y value is 6, as the 0 before the final "11111" is at position 6.

My question is: which distribution does the Y value have, if for each single trial the probability of success equals p?

BenP
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  • https://en.wikipedia.org/wiki/Negative_binomial_distribution. For relevant posts here on CV, see this site search. – whuber Mar 01 '24 at 17:39
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    @whuber: the negbin was also my first thought. However, this is not about $k$ successes (or failures), but $k$ successes in a row. That changes matters. – Stephan Kolassa Mar 01 '24 at 17:47
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    Thanks: I overlooked that criterion. You can find helpful information at https://stats.stackexchange.com/questions/12174 and https://stats.stackexchange.com/questions/544811/. And of course you can formulate it as a Markov chain and apply the usual algebraic machinery. – whuber Mar 01 '24 at 17:48

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