Odds question from a newbie

Ok Let's say there have been 48 championship games in a sport that consists of two 16 team divisions. What are the odds of one team only going once and losing?

Quote: james83

Ok Let's say there have been 48 championship games in a sport that consists of two 16 team divisions. What are the odds of one team only going once and losing?

How much do they suck?

If you pick a specific team, and assume everyone is equally skilled, then they will make it one time in 48 with probability:

48*(1/16)*(15/16)^47 = 14.4%

and will lose 50% of the time, so 7.2%.

If you want to know the probability of this happening to any team, that takes more work.

Let's say that's already happened. This particular team only made one appearance and they lost it. And now we want to figure what the odds of that was. Wound 7.2% still be the answer or is it more complex than that.

And yes all things are equal, we'll just say the sport is a coin flipping competition or something.

Quote: james83

all things are equal, we'll just say the sport is a coin flipping competition or something.

Coin flipping is beatable, and is certainly not fair and random.

Quote: Dieter

Coin flipping is beatable

Apparently so is this guy's favorite team.

Quote: AxiomOfChoice

Apparently so is this guy's favorite team.

That's okay, the Lions have had 48 attempts and not even made it to the big game. And they've played in season with less than 32 teams...

Quote: thecesspit

That's okay, the Lions have had 48 attempts and not even made it to the big game. And they've played in season with less than 32 teams...

The team I'm referring to is the falcons, they have been once and lost. I know there use to be fewer teams but I left that out to simplify the equation. I have a friend who claims that for a team to only have been once and lost requires incredible odds. Now I don't know what the odds are But I don't believe they would be "incredible" at all. I just can't wrap my head around how to calculate the exact odds and its driving me nuts.

Quote: thecesspit

That's okay, the Lions have had 48 attempts and not even made it to the big game. And they've played in season with less than 32 teams...

They must really suck at flipping coins.

Either that, or football.

Sadly not really.

For any particular team, considering they are all even, the chance of making the bowl is 1 in 16.

So for any year it's 15 in 16 they don't make it.

For two years, 15 in 16 times 15 in 16 = 225 in 256 or 87.9%.

For 48 years: 15 in 16 to the power of 48 : 4.5% they never get there.

As they go there exactly once, it's 1 in 16 times 15 in 16 to power of 47 , times 48 (48 different arrangements of the ordering) = 14.4%

So it's 14.4% chance that YOUR team will go to the superbowl ONCE is 48 years. Consider the final to be a coin flip... It's 7.2% chance they go once and lose. Therefore there should be (on average) 2.3 teams in this situtation.

There's actually 5 - Falcons, Panthers, Cardinals, Titans and Chargers. 4 have never made it (Browns, Lions, Jaguars, Texans). The Bills and Vikings are 0-4 and the Eagles and Bengals are 0-2. I -suspect- the assumption that all teams are equal over the 48 years is not correct (suspect... I reckon I could prove it) so it'd be even greater chance that a team would be 0-1 in 48 years.

I'd rather be a Falcon than a Bill....

Quote: AxiomOfChoice

They must really suck at flipping coins.

Either that, or football.

Both. If I'd known the night I stayed up to watch Sanders tear apart the Cowboys defence to give them the first ever playoff win was going to be the only one in 22 years... I'd have started following else.

Quote: thecesspit

Sadly not really.

For any particular team, considering they are all even, the chance of making the bowl is 1 in 16.

So for any year it's 15 in 16 they don't make it.

For two years, 15 in 16 times 15 in 16 = 225 in 256 or 87.9%.

For 48 years: 15 in 16 to the power of 48 : 4.5% they never get there.

As they go there exactly once, it's 1 in 16 times 15 in 16 to power of 47 , times 48 (48 different arrangements of the ordering) = 14.4%

So it's 14.4% chance that YOUR team will go to the superbowl ONCE is 48 years. Consider the final to be a coin flip... It's 7.2% chance they go once and lose. Therefore there should be (on average) 2.3 teams in this situtation.

There's actually 5 - Falcons, Panthers, Cardinals, Titans and Chargers. 4 have never made it (Browns, Lions, Jaguars, Texans). The Bills and Vikings are 0-4 and the Eagles and Bengals are 0-2. I -suspect- the assumption that all teams are equal over the 48 years is not correct (suspect... I reckon I could prove it) so it'd be even greater chance that a team would be 0-1 in 48 years.

I'd rather be a Falcon than a Bill....

Awesome man! Big thanks to you and dwheatley for breaking that down for me.

I think you have to take the odds one step further, though. There's a huge emotional component to the Superbowl that's not felt by cards, dice, other things where odds can be calculated. The Wizard probably has the breakdown at his fingertips, but there's a definite and well-remarked bias towards teams that have been there before (within the experience of that general squad). There's just such a build-up of expectation and hype that compares to nothing in a teams' prior experience, it influences their chance of competing with an equal chance of winning once they get there.

Quote: beachbumbabs

I think you have to take the odds one step further, though. There's a huge emotional component to the Superbowl that's not felt by cards, dice, other things where odds can be calculated. The Wizard probably has the breakdown at his fingertips, but there's a definite and well-remarked bias towards teams that have been there before (within the experience of that general squad). There's just such a build-up of expectation and hype that compares to nothing in a teams' prior experience, it influences their chance of competing with an equal chance of winning once they get there.

The Wizard probably has an even better way of doing it than I do, but what I would do is look up the Moneyline or Spread on the game in question:

http://en.wikipedia.org/wiki/Super_Bowl_XXXIII

In this case, we see the Broncos were a 7.5 point favorite, so using this chart (halfway down the page)

http://techguyinmidtown.com/2009/09/25/probability-nfl-favorite-wins-given-point-spread/

The spread implies a 69.11% probability of a Broncos win.

Therefore, the implied probability of the Falcons winning the game was 30.89%, thus, we take the .144 (probability of only appearing once) times .6911 (probability of losing) and get:

.144 * .6911 = .0995184 or a 9.95184% probability of this situation happening to the Falcons.

Of course, we can only do this because we know they played the Broncos, from an a priori standpoint, I think it would be difficult to mathematically quantify anything other than the general because we have no way of knowing who the opponent would be.

Is this unusual or unexpected? Yes and no. If the general probability of going once and losing is .072, then the probability of not being so situated is .928, but then the probability of none of the thirty-two teams being so situated is (.928)^32 = 0.09152405196 which is less than the specific team, the Falcons being so situated (now that we know they played the Broncos) and only slightly greater than a specific team being so situated, assuming the SB is a coin-flip.

On the flip side of this coin you have teams such as the New England Patriots, who won three out of four Super Bowls, which to have that happen to a specific team on an a priori basis has to be far less likely than the Falcons being situated the way they are.

It's kind of like Craps, in a way. If you have 20 rolls on a Craps Table, regardless of the results, that specific series of 20 results is going to be highly unlikely...but something MUST happen. This is essentially the same thing, you're looking at a series of many events spanning many years and picking out a specific scenario (already knowing what it is) that doesn't seem likely. What would be even less likely is for all thirty-two teams to have won at least one SB and to have won no more than two, even though, on an individual basis, those should technically be the, "Most likely," results.

Quote: thecesspit

That's okay, the Lions have had 48 attempts and not even made it to the big game. And they've played in season with less than 32 teams...

Yep, and you have to go back to the John Henry Johnson era, 1957, to their last championship. There were only 12 teams in the NFL then.

Quote: chickenman

Yep, and you have to go back to the John Henry Johnson era, 1957, to their last championship. There were only 12 teams in the NFL then.

Yes. Thanks. That's like me standing in the ocean and you telling me I'm getting wet :)

Quote: Mission146

It's kind of like Craps, in a way. If you have 20 rolls on a Craps Table, regardless of the results, that specific series of 20 results is going to be highly unlikely...but something MUST happen. This is essentially the same thing, you're looking at a series of many events spanning many years and picking out a specific scenario (already knowing what it is) that doesn't seem likely. What would be even less likely is for all thirty-two teams to have won at least one SB and to have won no more than two, even though, on an individual basis, those should technically be the, "Most likely," results.

Given teams are all equal over 48 years, it's odds on that one team (at least) is 0-1 in superbowls. Given that it's almost certainly not the case that all teams are equal over 48 years, it's even more likely that any one team you pick will go 0-0 in superbowls over the -next- 48 years. I'm not quite sure if the inequality makes 0-1 more or less likely, though.