Probability of multiple occurences over x amount of tries?

Question

As I understand, if probability of event is 3%, over 100 attempts probability of that event occuring is 1 - (0.97^100) = .9524. If that's correct?

Please advise what equation could be used to look at the odds of that event occuring x amount of times (i.e. at least twice) in 100 attempts?

Edit: sorry I'm having as much difficulty explaining as I am wrapping my head around it. These would be independent events each time. 3% chance instance 1, 3% instance 2, and so on. So i.e. how could I consider the odds that that event happened y times over x instances?

Thank you!

Answer 1

Yes, to part one. So you are on the right track.

That works because the chances of the thing not occurring is 97%. And, if you do it 100 times and they are independent, the chances of it never occurring are $.97^{100}$. So the chances of it happening at least once (which are the chances of it not never occurring) are one minus that.

It gets more challenging when you try to calculate the odds of it happening exactly N times out of M trials.

See if you can work this out. Assume you flip a coin 3 times. What are (1) the chances of no heads; (2) the chances of 1 head; (3) the chances 2 heads; and (4) the chances of three heads.

Hint: there are 8 ways a toss of a coin 3 times can come out - write them all down (like HHH, HTH, THT, etc.). That will get you started.