How to Calculate the Geometric Distribution

This is the distribution of how many attempts it takes to succeed for the first time: $k-1$ failures followed by a success on attempt $k$.

The chance of succeeding within $k$ attempts is the opposite of never succeeding, $1 - (1-p)^k$. The mean is $1 \div p$ and the standard deviation $\sqrt{1-p} \div p$.

Example

With a 20% chance each time, the first success falls exactly on the third attempt with probability $0.8^2 \times 0.2 = 0.128$, or 12.8%. The chance of succeeding within three attempts is $1 - 0.8^3 = 0.488$, or 48.8%. The mean is $1 \div 0.2 = 5$ attempts and the standard deviation 4.47.

Waiting for the mean is not enough

Even with a mean of 5, the chance of succeeding within 5 attempts is only $1 - 0.8^5 = 67.2$%. Roughly one time in three, five attempts still come up empty. The distribution has a long tail to the right, which pulls the mean above the median.

It has no memory

However many attempts have failed, the next one still succeeds with probability $p$. Nothing is due. This is called memorylessness, and the exponential distribution has the same property in continuous time.

Notes

With a success probability of 0 there is never a success, so no mean exists.