What is the probability of a team not playing an a certain field

Question

There are 12 teams in a league that play 16 games each per season. There are 3 possible fields to play on each week (16 games total). What is the probability that only one team does not get to play on field 1? There is a field 1, field 2, and field 3 available for all teams to play on.

Answer 1

This is an example of poorly worded (or misworded) problem. So let us first solve see the intentions of problem writer.

Firstly problem writer has hidden assumptions

1. each game is between two teams and that each game
2. each game happens on random field, independently

Then intention was as follows.

Then define event

E_i as team i not playing on field 1 and

E as probability that only one team did not play on field 1

P[A_i] = P[one game is on field 2 or 3]¹⁶ = (2/3)¹⁶

~= 0.001523, call this p₁

Then assuming E_i are independent one gets

P[E] = 12 * p₁* (1-p₁)¹¹

~=0.01797

The key reason why this is a terrible problem is that the error in the reasoning above is, in almost any thinkable setup events E_i are not independent and then it becomes even bigger mess. However apprximate numerical answer should be close because events like two teams did not play on field 1 are already a lower order term and their correlation correction would be another order of magnitude smaller, so the answer should be correct to first 2 or 3 digits in the least.