The answer here is likely, “No, you do not have enough information.”
Considering the NCAA March Madness tournament, what I would like to do is, starting with pre-tournament probabilities for Team T, winning its Nth round game: P(T,N), how may I update this once I know the first round’s results?
Prior to the tournament start we have probabilities of each team winning their game in Round 1 and Round 2. P(T,N) will represent the probability of team T winning its Nth round game. Thus P(2,1) is the probability of Team 2 winning its first round game.
To give an example of one 4 team sub-bracket’s first two rounds:
P(1,1) = 75, P(1,2) =36, P(2,1) = 25, P(2,2) = 7, P(3,1) = 81, P(3.2) = 50, P(4.1) = 19, P(4,2) = 7
Since I am only looking at the second round, the updated second round probabilities can only possibly be affected by the results of the four teams in the local bracket.
First round: Game 1-Team 1 plays Team 2, Game 2 -Team 3 plays Team 4
Second round: the Winner of Game 1 plays the winner of Game 2.
Now assume the first round games are played and Team 1 and Team 4 win.
Obviously: P(2,2) and P(3,2) are now 0. What are the new P(1,2) and P(4,2)? Assume no other information is gathered other than the outcome of the first 2 round games,
I was hoping an application of Bayes' Theorem would work here. A - the event Team 1 wins Game 2. B - the event Team 4 wins Game 1. Then we are looking for P(A|B) = (P(B|A) * P(A))/P(B). We would have P(A) and P(B), but I could not figure out P(B|A) either..
Is this possible? Other ideas?
Considering the NCAA March Madness tournament, what I would like to do is, starting with pre-tournament probabilities for Team T, winning its Nth round game: P(T,N), how may I update this once I know the first round’s results?
Prior to the tournament start we have probabilities of each team winning their game in Round 1 and Round 2. P(T,N) will represent the probability of team T winning its Nth round game. Thus P(2,1) is the probability of Team 2 winning its first round game.
To give an example of one 4 team sub-bracket’s first two rounds:
P(1,1) = 75, P(1,2) =36, P(2,1) = 25, P(2,2) = 7, P(3,1) = 81, P(3.2) = 50, P(4.1) = 19, P(4,2) = 7
Since I am only looking at the second round, the updated second round probabilities can only possibly be affected by the results of the four teams in the local bracket.
First round: Game 1-Team 1 plays Team 2, Game 2 -Team 3 plays Team 4
Second round: the Winner of Game 1 plays the winner of Game 2.
Now assume the first round games are played and Team 1 and Team 4 win.
Obviously: P(2,2) and P(3,2) are now 0. What are the new P(1,2) and P(4,2)? Assume no other information is gathered other than the outcome of the first 2 round games,
I was hoping an application of Bayes' Theorem would work here. A - the event Team 1 wins Game 2. B - the event Team 4 wins Game 1. Then we are looking for P(A|B) = (P(B|A) * P(A))/P(B). We would have P(A) and P(B), but I could not figure out P(B|A) either..
Is this possible? Other ideas?