Once again, I'm coming to Brilliant with another question. This time, it's on the limits of probabilities.

I was investigating this problem:

You are playing a computer game with \(n-1\) of your friends (a total of \(n\) people), and each of you get a chest with a \(\frac{1}{n}\) chance of giving you a prize. What is the probability that none of you get a prize?

I realized that each party member has a \(\frac{n-1}{n}\) chance of not getting a prize. This approaches 100% as you have more and more party members. The party probability is easy to find once you find the individual probability. Since the probability for each party member is \(\frac{n-1}{n}\), you just need to raise it to the \(n^{\text{th}}\) power, since there are \(n\) people in the party.

Then, I got curious about the infinite limit of the function: \[\displaystyle\lim_{x\rightarrow \infty} f(x)=\left(\frac{x-1}{x}\right)^x\] After I graphed it, I realized that there is a horizontal asymptote, and it cannot be calculated through traditional methods. Could someone explain how to get the solution (not just through WolframAlpha...)? I know it is approximately \(y=.368\) through direct observation.

*It is actually very interesting that, if there is an arbitrarily large amount of people, the probability that no one will get a prize is more than a third! Talk about bad odds!*

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## Comments

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TopNewestWell, since the probability of a person getting a prize depends on the number of people themselves, the result is not so surprising.

How did you stumble upon this problem?

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Actually, this problem is based off of an actual scenario that I was involved in. I just got curious and decided to generalize it.

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Did you realise that the number you produced is actually \(\frac{1}{e}\)? You have a real scenario? That is interesting. Can you tell me about it?

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Scenario: I play this mobile game called Clash Royale. They recently added a feature where everyone in your group, called a clan, gets a chest after winning in an event. The winning chest has a 1 in 10 chance of giving you a special prize. Now, my clan has 30 members, so I knew that I should expect 3 people in my clan to get the prize. Then, I started getting curious about the probability ofno onegetting a prize. You can infer what happened from there: I calculated it, decided to generalize for a clan of \(n\) members with a 1 in \(n\) chance of getting the prize, and then I got here.Log in to reply

I am not sure I am following. Why does the chance of someone getting a prize in the clan reduce when there are more people?

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In the original scenario, the probability of getting the prize was independent of the number of members in the clan. I modified that scenario, giving us the one mentioned in the note.

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Cool. Maybe you can post this as a problem?

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Here it is! Thanks for the help!

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