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.

 

[tacman]

Hello, thank you for reaching out for help with percentages when stacked or accumulated. Calculating probabilities and odds can be a bit tricky, but I will do my best to explain it in a clear way.

First, let's define some terms. Probability is the likelihood of an event occurring, expressed as a number between 0 and 1. For example, a 20% chance of rolling a 1 on a 5-sided dice would be written as 0.2. Odds, on the other hand, are expressed as a ratio or fraction of the chances of an event happening compared to the chances of it not happening. In our example, the odds of rolling a 1 on a 5-sided dice would be 1 in 4, or 1/4.

To find the probability of an event happening multiple times in a row, you can simply multiply the probabilities of each individual event. For instance, if you want to know the probability of rolling a 1 twice in a row on a 5-sided dice, you would multiply 0.2 by 0.2, which equals 0.04 or 4%. Similarly, for three times in a row, you would multiply 0.2 by 0.2 by 0.2, which equals 0.008 or 0.8%.

To calculate the odds of an event happening at least one time in a certain number of attempts, you can use the formula 1 - (1 - p)^n. In this formula, p represents the probability of the event happening, and n represents the number of attempts. For example, if you want to know the odds of rolling a 1 at least once in 15 attempts on a 5-sided dice, you would plug in 0.2 for p and 15 for n. The equation would look like this: 1 - (1 - 0.2)^15 = 0.965 or 96.5%. This means that there is a 96.5% chance of rolling a 1 at least once in 15 attempts.

Lastly, to address your question about how the chances increase with each roll, the probability of an event happening does not change based on previous outcomes. Each roll of a dice or click of a button is an independent event, meaning that the outcome of one does not affect the outcome of the next. Therefore, the probability of rolling a 1 on a 5-sided