What are the odds of this occurring?

Question

There are numbers from 1 to 2500.
There are 625,000 people.
What are the chances that 3 people pick the same 4 numbers?
The order does not matter and the people cannot repeat.

Answer 1

There are C(2500,4) possible ways to select 4 numbers from 2500.

Then, for a given combination, the probability that 3 of 625000 will select it is

C(625000,3) (1/C(2500,4))^3 (1-1/C(2500,4))^625000-3

Then, as this is the probability for one number, the probability for all of the numbers is approximately

C(2500,4) C(625000,3) (1/C(2500,4))^3 (1-1/C(2500,4))^(625000-3) = 1.54 * 10^-8

 

(You could also make it 1 - (1-C(625000,3) (1/C(2500,4))^3 (1-1/C(2500,4))^(625000-3))^C(2500,4); these numbers are very similar.