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In a video game I'm playing, there is a random number generator that awards one trophy from a pool of possible trophies. The pool of possible trophies is approximately 60 different items, and I was supposed to be awarded 5 random ones. I received 3 of one type, and 2 of another.

I'm trying to estimate the odds of this occurrence, versus the odds of receiving 5 different ones.

For the "3 of the same type" the odds would be 1 out of 3600, and the odds of "2 of the same type" is 1 out 60. Or not?

Does the order of events impact the odds? If so, the order was: #1, #1, #2, #1, #2. Would this imply the odds at each stage were NA, 1:50, 1:49, 1:48, 1:48 ?

Apologies in advance if my terminology is incorrect. I know "odds" and "probability" are not interchangeable, but it's been a LOOOONG time since my college statistics class.

TIA.

software evolved
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  • You appear to be assuming that each award is equally probable. Is that the case? Often in these sort of situations, some awards are "special" and are rarely handed out while others are "ordinary" and are very likely to be awarded. Are you sure that's not the case here? – lulu Oct 23 '20 at 16:00
  • Do you mean that there are two different kinds of trophy, with a pool of 60 of each? So it's as if there's a bag of 60 white marbles and 60 black marbles, and you draw a random marble from the bag (without replacing it) when you get an award? – Théophile Oct 23 '20 at 16:01
  • Oh, but you ask about the odds of getting 5 different trophies. So how many kinds are there? 5? 100000? – Théophile Oct 23 '20 at 16:02
  • Sorry I wasn't clear. The game implies that each award (taken from a pool of 60 possible trophies) is independent of all previous awards and is equally probable. When a trophy is awarded it is NOT removed from the pool of possible trophies so there are always 60 possible, different trophies. Hence my surprise when, after 5 awards, I had only received 2 different types – software evolved Oct 23 '20 at 16:06
  • @softwareevolved Okay, that's clear now. The outcome you're describing is indeed surprising; maybe you were just very (un?)lucky, or, more likely, the trophy types are not equally distributed. – Théophile Oct 23 '20 at 16:21
  • Does this clarify my problem: "Given the numbers 1 to 60, what are the odds of selecting three of one number, and two of another, if five numbers are randomly selected each time?" – software evolved Oct 23 '20 at 16:23

1 Answers1

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If the reward system is truly random, the order does not matter. The probability of getting any award is $1/60$.

Now your question depends on one thing: it matters if you ask ''what is the probability of this specific situation happening'' or ''what is the probability that this situation, or any other ''comparable'' situation'' happens.

The difference I'm pointing at is whether you want to know the probability of the awards you got (that would be $(\frac{1}{60})^5)$, but I think you mean any situation in which you have 3 of one type and 2 of another type.

In that case, it's different. There are $60*59= 3540$ ways to choose the two types of trophies you get. The probability of each one of those ways is still $(\frac{1}{60})^5$, so the overall probability is $3540*(\frac{1}{60})^5$ which is around $0.00046\%$ or, very roughly, about $1$ in $220\ 000$.

Edit: Note that the probability of getting $2$ types of trophies from the $5$ you got, disregarding how many you got of either, is twice as high, but still very small. This is because you can then have either a $2+3$ or $1+4$ combination, so that gives twice as many options.

Marc
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