Help with percentages when stacked/accumulated

[Spero]

Hello,
I was hoping someone would help me with percentages/odds when multiplied. I will use dice and buttons just to make my explanation slightly easier to understand.

Example: A 5 sided dice would have a 20% chance or 1 in 5 of getting a 1 (or any other number), what I want to know is how to find out the odds of getting a 1 once if it was rolled 2 times...5 times...9 times etc. Or a button that you click, has a 12% chance of giving you the message HELLO! How would you figure out the chances of it saying HELLO! at least one time if you clicked it 15 times or 35 times etc.

Obvously if you roll that 5 sided dice 5 times it doesn't mean you have a 100% chance of rolling a number. I know this is also random but is there some equation to figure out how much the chances increase with each roll? I'd think first time would be 20% you get a 1, 2 rolls give like a 30%, 3 rolls like a 35% or something. Hopefully this is clear enough so people know what I'm asking. If not tell me what I should clarify.

Thanks for any help in advance on this.

 

[Tide]

First, you need to decide whether you want to calculate probability or odds. They are not the same thing (odds = a ratio of proabilities). Assuming you mean probability then on any given toss you have 1 chance in 5 of rolling a particular specified number but 4 chances in 5 of not getting that number.

With n tosses, the probability of exactly one of those tosses being the particular specified number is

C(n, 1) x (1/5) x (4/5)^(n-1)

which is the same as tossing n dice all at once and having exactly one of them be the particular specified number. C(n, r) accounts for the fact that you are "choosing" one of the dice to be the specified number out of a collection of n dice.