# Geometric distribution with multiple trials

I was looking into geometric distributions to find the probability of the first success of some random variable X. So if p = 0.04, the geometric distribution looks something like this:

I understand the graph mathematically from the probability formula, but it seems kind of unintuitive for the chance of first success being x to keep going down, since I wouldn't expect getting a success on the first try to have the highest probability with p = 0.04.

So if I run an experiment with a random number generator with chance of success = 0.04, and record the trial in which I get my first success and plot it on a graph, would it be expected to look something like the above distribution, where getting a success on the first try would be the most common event?

• Probability is strange, so we need to calibrate our intuitions by analyzing simple situations like this. Consider that if you don't have a success on the first try, you're back where you began. This happens 96% of the time, so as a result your chance of a success on the second overall try is 96% of the chance of getting a success on the first try, etc.
– whuber
Commented Jul 16, 2019 at 22:42