How can I calculate the probability that at least 2 people out of $k$ people choose an identical set of $n$ numbers? The numbers are positive integers, starting with $1$, and every number only appears once. We assume that every person picks their numbers randomly.

Example: $3$ people choose $5$ numbers between $1$ and $20$:

  • Person 1: $5, 8, 10, 12, 13$
  • Person 2: $4, 5, 6, 7, 8$
  • Person 3: $5, 8, 10, 12, 13$

How likely is it that Person 1 and Person 3 picked the same numbers?

I would need the formula in a generic way so that $k$ and $n$ can vary. Any help is appreciated! Bonus: How would I need to modify the formula if I want to know the probability that "at least 3 people" choose the same numbers?

  • $\begingroup$ Are the numbers ordered? Is there a difference between picking $1,2,3,4,5$ and $5,4,3,2,1$? $\endgroup$ – Kevin Long Oct 21 '16 at 15:13
  • $\begingroup$ no, there would be no difference, the order is irrelevant $\endgroup$ – user2037036 Oct 22 '16 at 12:37

Hint: The probability that at least two people pick the same set of numbers is equal to $1-$ the probability that everybody picks different sets of numbers.

  • $\begingroup$ Thanks, so how can I calculate the probability that everybody picks different sets of numbers? Unfortunately my math skills are not very sophisticated... $\endgroup$ – user2037036 Oct 22 '16 at 12:38
  • $\begingroup$ @user2037036 Let's say the first person picks $k$ numbers. Then the second person can pick $k$ out of $n-k$ numbers. Do you know how to count this? $\endgroup$ – Kevin Long Oct 22 '16 at 15:40

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