Probability of drawing different items from a subset with replacement

Question

I'm trying to generalize a game mechanic, to understand the probability that the game developers are giving us, but this went over my head.

The mechanic is:

• draw 10 items from a bag of 33 items
• each draw is independent (with replacement)
• 5 of the items are "badges"
• you win if 3 different badges are drawn

Thank you!

Answer 1

I am going to assume that your question is "what is the probability of winning this game", that is drawing exactly 3 different badges. A simple simulation will get you your result.

> bag=c(rep(0,28),1:5)
> rez=replicate(1e5,{
>   smp=sample(bag,10,replace=T)
>   return(length(unique(smp[smp!=0])))
> })
> 
> prop.table(table(rez))

rez
      0       1       2       3       4       5 
0.19221 0.40511 0.29756 0.09257 0.01201 0.00054

So the probability of "winning" is roughly $9.3 \%$.