The geometric distribution. Generalize your work in the previous exercise. You have independent trials, each resulting in a success or a failure. The probability of a success is p on each trial. The binomial distribution describes the count of successes in a fixed number of trials. Now the number of trials is not fixed.
Instead, continue until you get a success. The random variable Y is the number of the trial on which the first success occurs. What are the possible values of Y?
What is the probability P(Y = k) for any of these values? (Comment: The distribution of the number of trials to the first success is called a geometric distribution.)