The condition was satisfied.
There are a few things to parse here. Use words very carefully and things will be clear.
First, the probability p is a number between 0 and 1 inclusive. It doesn't occur or happen, it just is.
Your computer is generating a (likely random) number between 0 and 1 inclusive, and testing whether the generated number is less than the probability p. In this case you would say the condition was satisfied.
If the condition is satisfied, you consider that an event, such as the triggering of an action, or logging that the condition was satisfied. In the former case you can say the action was triggered, or that the event occurred.
You could say the X event occurred, or that the X condition was satisfied, after establishing that X is the event or condition associated with probability p. In no case would you state p by name, unless you had a very wordy the condition associated with p was satisfied. In your case if you'd like to use "trigger", then the condition associated with p was triggered would work.