If we are interested in calculating the probability that a first success occurs on the 6th trial, what do we know about the outcomes of the first 5 trials?

1 Answer
Jeffrey W.
Jan 1, 2015

The first 5 trials must not be successes.

To calculate the probability of an event with probability p first occurring on the sixth trial, one must assume that the event fails 5 times and then succeeds. Because each of the trials in a geometric distribution are independent, identically distributed binary events, we can simply multiply each of the 5 non-successes and the 1 success.

Because the probability of an event that succeeds with probability p not succeeding is (1 - p), the probability of the event first occurring on the sixth trial is equal to:

(1-p)^5*p

In fact, the probability of the event occurring on the n^"th" trial is:

(1-p)^(n-1)*p

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