interesting AI interaction

I asked the free version of Gemini a mathematical question pertinent to my gambling career. It's a combinatorics question and not that difficult.

1st I just asked for the odds of two weakly related events to happen.

AI came back and said: I used a method that gives a good approximation of the answer and it is xxxx.

2nd version of the question I specified I wanted the EXACT answer to the same question.

AI came back and said: I used the binomial probability method that gives a very good approximation of xxxxx. The answer will be very close because the two events are only weakly related.

3rd version of the question was same as the 2nd but on the end I added DO NOT USE THE BINOMIAL PROBABILITY method.

AI came back and said: The binomial probability method gives xxxx as the answer. The combinatorics method gives the precise answer and it is yyyyyy.

I found it kind of funny, actually. I was surprised that Gemini understood the cause of the inaccuracy of the shortcut method and knew all the relevant details required despite me not including those in my question.

FYI: the shortcut method is off by 30%, so to me that's not a good approximation.

Edited May 4May 4 by gamb00ler