Bias and Standard Deviation of the Estimator for the Underlying Probability in a Sequence of Bernoulli Trials with Preselected Number of Successes

In a sequence of Bernoulli trials which stops after a preselected number M of successes, the maximum likelihood estimator of the underlying probability is given by the ratio of M and the required number of observations. Closed-form expressions for the bias and standard deviation of this estimator are developed.