Question

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?

Answer 1

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`