Independent Bernoulli trials are performed, with probability $1/2$ of success, until there has been at least one success. Find the PMF of the number of trials performed.
How is this different from the negative binomial?
[Math] Bernoulli trials with at least 1 success and 1 failure
probability distributions
Best Answer
This is the geometric distribution, # of trials needed to get first success
$P(x=k) = (1-p)^{k-1}\cdot p$
The negative binomial distribution gives the # of trials needed to get k successes
So, as has been commented, the geometric distribution is a special case of the negative binomial distribution.
Note
Some count the # of failures for the negative binomial, but as you must be knowing, what is defined as "success" is just a matter of convenience.