Voodoo

[Math Demon]

@
For Mr. Tthree and other mathematically-inclined members:


Let's assume that there is a hypothetical NFL team, New Orleans Voodoo. http://en.wikipedia.org/wiki/New_Orleans_VooDoo

For 20 NFL seasons, through various GMs, coaches and players, this team (using a secret voodoo ritual@ ) had managed a unique feat -- the team NEVER loses two games in a row. They could win 16 consecutive games, but they never lose 2 consecutive games.


For simplicity, let's further assume:

1. Each NFL season consists of 16 regular season games, and up to 4 playoff games.
2. After a win (regular and post-season), the next game has a 50% win probability. After a loss (regular season), the next game has a 100% win probability.
3. The team's streak continues from regular season to post-season, i.e., if they lose their last regular season game, the chances for the first post-season game win is 100%.
4. Pre-season results are ignored. The first regular season game has a 50% win probability.
5. This team's lowest record ever is 8-8 (Duh!) @ In such a case, there is no playoff appearance for them that year.
6. If this team ends with 10-6 or better, they win the Division, and plays only 3 post-season games (Div, Conf, SB).
7. If this team ends with 9-7: half-the-time, they go home, half-the-time, they play as Wild Card with 4 post-season games (WC, Div, Conf, SB).


If the pattern holds ... in any given year, what is the probability that Voodoo will:

1. Go 8-8 and miss the playoffs?
2. Go 9-7 and miss the playoffs?
3. Get in the playoffs as "Wild Card" (50% of 9-7)? And win on that round? Lose that round?
4. Get in the playoffs as "Div Leader" (10-6 or better record)? And win on that round? Lose that round?
5. Win their Conference?
6. Win the SuperBowl?

7. How do the probabilities #1 to #6 change depending on whether they win or lose their 1st regular season game?



Thanks,
MD



Addendum: Maybe a further simplification could be to lump the wins together ... thereby reducing the 65,536 win-loss matrix into 8 wins plus 256 win-loss matrix?
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[Three]

If the pattern holds ... in any given year, what is the probability that Voodoo will:

1. Go 8-8 and miss the playoffs?
2. Go 9-7 and miss the playoffs?
3. Get in the playoffs as "Wild Card" (50% of 9-7)? And win on that round? Lose that round?
4. Get in the playoffs as "Div Leader" (10-6 or better record)? And win on that round? Lose that round?

1) 9/2584
2) 60/2584
3) 60/2584; 7,200/403,104; 2,160/403,403
4) 2455/2584; 1,134,210/1,736,448; 515,550/1,736,448

[Math Demon]

1) 9/2584
2) 60/2584
3) 60/2584; 7,200/403,104; 2,160/403,403
4) 2455/2584; 1,134,210/1,736,448; 515,550/1,736,448

Thanks very much, Tthree!

But can you elaborate on the approach you took to derive the numbers? Did you proceed via "Bernoulli trails" with binomial distribution? Thanks again!

[Three]

If you break down the winners and losers at each record as the season plays out, A record of 6-2 would have the entire 5-2 group having won the last game and the only the 6-1 contributing losers from those that had won the game in week 5. Basically there will be a fibonacci sequence for the number of possibilities each week. Keep each week as a number pair for each record by won last game and lost last game. Take the sum of the pair from the same number of losses in the previous week as the winning part of this weeks pair and take the winning part of the pair from last weeks pair for the same number of wins. The total of any pair is the number of combinations to make that record and the fibanocci number corresponding to that week is the total number of combinations for that week. I wasn't sure about using bernouli trails were the probabilities were dependent on the last results so I took another approach that is similar but easier to visualize.

[Math Demon]

If you break down the winners and losers at each record as the season plays out, A record of 6-2 would have the entire 5-2 group having won the last game and the only the 6-1 contributing losers from those that had won the game in week 5. Basically there will be a fibonacci sequence for the number of possibilities each week. Keep each week as a number pair for each record by won last game and lost last game. Take the sum of the pair from the same number of losses in the previous week as the winning part of this weeks pair and take the winning part of the pair from last weeks pair for the same number of wins. The total of any pair is the number of combinations to make that record and the fibanocci number corresponding to that week is the total number of combinations for that week. I wasn't sure about using bernouli trails were the probabilities were dependent on the last results so I took another approach that is similar but easier to visualize.

Truly, you have a dizzying intellect, Mr. Tthree!

It took me some time to understand what you stated. And it took modifying the Bernoulli trials with the logical constraints for the Fibonacci pattern to appear.

I concur that there would be 2584 possibilities after game 16. However, I still think that the team would be 8-8 only 2/2584 or 0.08% -- with the two possibilities coming from alternating wins and losses after an initial win or loss on game 1.

I'll look into the other subsets (9-7, 10 or better) later tonight.
__

Having said that, if any NFL team can manage to prevent consecutive losses, they would never finish below 8-8. They would finish 8-8 only 1/10 of 1% of the time. And the mode, median and mean of their season wins would be 12 games -- certainly enough to win a Division and a possible first-round bye.


Thanks again!

[Math Demon]

If the pattern holds ... in any given year, what is the probability that Voodoo will:

1. Go 8-8 and miss the playoffs?
2. Go 9-7 and miss the playoffs?
3. Get in the playoffs as "Wild Card" (50% of 9-7)? And win on that round? Lose that round?
4. Get in the playoffs as "Div Leader" (10-6 or better record)? And win on that round? Lose that round?


1) 9/2584
2) 60/2584
3) 60/2584; 7,200/403,104; 2,160/403,403
4) 2455/2584; 1,134,210/1,736,448; 515,550/1,736,448

Thanks, Tthree!

Your numbers are exact and correct. Your approach to the problem was beautiful!