What is the probability?

Question

Once a week, folks from Blacksburg, Greensboro, and Silver Spring get together for a game of pickup basketball. Every week, anywhere from one to five individuals will show up from each town, with each outcome equally likely.



Using all the players that show up, they want to create exactly two teams of equal size. Being a prideful bunch, everyone wears a jersey that matches the color mentioned in the name of their city. However, since it might create confusion to have one jersey playing for both sides, they agree that the residents of two towns will combine forces to play against the third town’s residents.

What is the probability that, on any given week, it’s possible to form two equal teams with everyone playing, where two towns are pitted against the third?

Extra credit: Suppose that, instead of anywhere from one to five individuals per town, anywhere from one to N individuals show up per town. Now what’s the probability that there will be two equal teams?

Answer 1

Because one side must equal the sum of the other 2, we will have 3 possible large groups of size 2, 3, 4, and 5. The two other two towns can sum to 2 in 1 way, 3 in 2 ways, 4 in 3 ways, and 5 in 4 ways.

Thus, 3 * (1+2+3+4) =3 * (4*(4+1)/2) = 3 * 10 = 30

30/125 = 6/25 = .24

As I calculated the sum by using a formula, it is now easy to generalize.

3 * N * (N-1)/(2N^3) = 3(N-1)/(2N^2)