Overtime Game Statistical Oddity

[SnowblindNYR]

I posted this in the "By the Numbers" board and was hoping to get feedback and haven't gotten any yet. I'd love to get feedback on whether or not my calculations are legit from someone who understands probability:

The Rangers have a statistical oddity this year. They played 27 games and have only played 1 OT game. I figured that this was quite an outlier and decided to do some probability (I'm a probability nerd) just to see how wacky it is. I would appreciate if a prob expert can confirm or deny my findings.

I added up all of the OT games in the NHL this year per team and games per team. Divided both by 2 (technically unnecessary) and got that in 405 games there have been 100 OT games. In other words just a little more rare than 1 every 4 games. Applying that to 27 games that the Rangers have played I got an expected 6.7 OT games (compared to 1). Now this is not a fair die or fair coin, so I can't really get a probability of games going into OT, but I decided to use approximately 24.7% as has been the case so far this season around the league.

I wanted to check to see what the probability of having a team go through 27 games and have OT "as rarely" as once in 27 games provided. Basically I did 1 or fewer OT games in 27. I first did OT in 0 games and that's (prob. of regulation finish in 1 game to the 27th power), then I did OT in 1 game (prob of regulation finish in 1 game to the 26th power multiplied by prob of OT in 1 game multiplied by 27 (27 scenarios where there's 1 OT in 27 games)). I got an amazing 0.004659665. Less than a half of a percent. That's a 0.995340335 chance that doesn't happen. I then took 0.995340335 and put it to the 100th power and subtracted the answer from 1 to see what are the chances that in 100 years of similar OT game paces that would happen. I got 0.373154691. In other words if the Rangers played 100 years there's only slightly more than a 37% chance that we see 1 or fewer games go to OT. In 150 years you finally get the odds in your favor and it's slim: 0.503704233. So just for it to be a little better than a coin flip chance of seeing such a scenario again you need to have 150 years worth of 27 game samples.

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[zuckera1]

I posted this in the "By the Numbers" board and was hoping to get feedback and haven't gotten any yet. I'd love to get feedback on whether or not my calculations are legit from someone who understands probability:

The Rangers have a statistical oddity this year. They played 27 games and have only played 1 OT game. I figured that this was quite an outlier and decided to do some probability (I'm a probability nerd) just to see how wacky it is. I would appreciate if a prob expert can confirm or deny my findings.

I added up all of the OT games in the NHL this year per team and games per team. Divided both by 2 (technically unnecessary) and got that in 405 games there have been 100 OT games. In other words just a little more rare than 1 every 4 games. Applying that to 27 games that the Rangers have played I got an expected 6.7 OT games (compared to 1). Now this is not a fair die or fair coin, so I can't really get a probability of games going into OT, but I decided to use approximately 24.7% as has been the case so far this season around the league.

I wanted to check to see what the probability of having a team go through 27 games and have OT "as rarely" as once in 27 games provided. Basically I did 1 or fewer OT games in 27. I first did OT in 0 games and that's (prob. of regulation finish in 1 game to the 27th power), then I did OT in 1 game (prob of regulation finish in 1 game to the 26th power multiplied by prob of OT in 1 game multiplied by 27 (27 scenarios where there's 1 OT in 27 games)). I got an amazing 0.004659665. Less than a half of a percent. That's a 0.995340335 chance that doesn't happen. I then took 0.995340335 and put it to the 100th power and subtracted the answer from 1 to see what are the chances that in 100 years of similar OT game paces that would happen. I got 0.373154691. In other words if the Rangers played 100 years there's only slightly more than a 37% chance that we see 1 or fewer games go to OT. In 150 years you finally get the odds in your favor and it's slim: 0.503704233. So just for it to be a little better than a coin flip chance of seeing such a scenario again you need to have 150 years worth of 27 game samples.

Great post, interesting findings. I wish I had as much free time as you do

 

[SnowblindNYR]

Great post, interesting findings. I wish I had as much free time as you do

Thanks. It took probably about a half hour (though I worked out the methodology beforehand), I'm at work, but I'm not exactly curing cancer here.

 

[Thirty One]

Another reason why I think the Rangers record is much worse than their play.

 

[SnowblindNYR]

Another reason why I think the Rangers record is much worse than their play.

Well they've won a bunch of close games too so they could have gone into OT in those rather than the losses. Though, I think the vast majority of our losses since the rough start were 1 goal losses, more than the 1 goal wins.

 

[SnowblindNYR]

BTW, I want to make a bit of a revision. I made it sound like by year 150 the odds turn in your favor. I didn't check that for sure. It's close to 150, but it's not necessarily that year. I took 150 because it's nice round number higher than 100. It luckily satisfies, both "round number" and "slightly over 50%" criteria.

 

[SnowblindNYR]

It'll take precisely 149 years to have odds turn in your favor (0.501380835). So 150 is damn close and a nice round number.

 

[SnowblindNYR]

LOL. I also want to explain that the years was taken for if we took 27 game samples a year. If we were to do it as 82 game samples, it would be 149 years divided by 3. About once ever 49 years. If you want to round up, if we took a regular season worth of games and took the same probability as used here (about 1 in 4), we'd see this happen a little more frequently than once in a half century.

 

[SnowblindNYR]

Ugh, that's probably not the correct way to say it. Basically the odds are in your favor that a team gets 1 of those 27 game samples in about 50 years and only slightly in your favor. Basically if you were to place a bet, the odds turn in your favor only after 50 years of these 27 game samples.

 

[sbjnyc]

Interesting observation. I'm an actuary but not necessarily an expert.

The real answer is that you have to consider this is only part of the season and there are 29 other NHL teams. So if the Rangers have a 0.5% chance of having exactly 1 game go to OT out of the 1st 27 games then perhaps there is a roughly a 15% chance of any NHL team having this occur (and maybe even a corresponding post on their board ). This is about 1 in 6 so it should occur every 6 seasons. The question to ask is how often has it actually occurred over the past 20 seasons or so. Statistics can then help determine whether this is a fluke or not.

But I'm not sure this is the real question. After all, there are 56 27 game spans during an 82 game season and 1,680 for all NHL teams (this ignores questions of independence since if 1 team plays more OT games than expected then the other teams they play will also play more OT games). In any case, if the chance for any one team to have a 27 game span with 1 OT game is 0.5% then you'd naively expect there to be 8-9 such events during a single season (expected values are additive).

As for the math, I think the mechanics are right but I think you're looking at too special a case - only the Rangers and only in the opening 27 game span. In 100 82 game seasons there are a total of 168,000 27 game spans. But you're only considering 100 of them (0.06%). IMO that's why it looks so odd.

 

[SnowblindNYR]

Interesting observation. I'm an actuary but not necessarily an expert.

The real answer is that you have to consider this is only part of the season and there are 29 other NHL teams. So if the Rangers have a 0.5% chance of having exactly 1 game go to OT out of the 1st 27 games then perhaps there is a roughly a 15% chance of any NHL team having this occur (and maybe even a corresponding post on their board ). This is about 1 in 6 so it should occur every 6 seasons. The question to ask is how often has it actually occurred over the past 20 seasons or so. Statistics can then help determine whether this is a fluke or not.

But I'm not sure this is the real question. After all, there are 56 27 game spans during an 82 game season and 1,680 for all NHL teams (this ignores questions of independence since if 1 team plays more OT games than expected then the other teams they play will also play more OT games). In any case, if the chance for any one team to have a 27 game span with 1 OT game is 0.5% then you'd naively expect there to be 8-9 such events during a single season (expected values are additive).

As for the math, I think the mechanics are right but I think you're looking at too special a case - only the Rangers and only in the opening 27 game span. In 100 82 game seasons there are a total of 168,000 27 game spans. But you're only considering 100 of them (0.06%). IMO that's why it looks so odd.

Actually you bring up good points. I only looked at one team. So if you're watching the Rangers for about 50 years you begin to more likely to get 1 or fewer. That said, I think what you're bringing in is that there are more than 3 27 game sample per team. There are actually 56 I think (if we're looking consecutively which I think makes more sense, if you're looking at a random sample of 27 then you have to look at the combination of 27 out of 82, which becomes an obscenely large number with 22 decimal digits). I guess you got 168,000 by multiplying 56 by 30 by 100. I don't like the 100 year sample, because it just randomly uses 1 27 year sample a year when there are way more. That's why I changed it to 3 per year, but you're right, there are far more than 3 per year.

If we're using ANY 27 game samples size across the league, the likelihood that it happens to a team in a season becomes 0.999608894. It happening to one team in a season, chances are 0.230143443. So it's pretty unlikely that it happens to a team in one season. However, the event occurring at some point during a season is extremely likely. Happening in the first 27 games is pretty irrelevant, it just stands out because no one looks at games 4-30.

Back to the drawing board. I guess I was humbled a bit.

 

[SnowblindNYR]

22 digits, not decimal digits, haha.

 

[SnowblindNYR]

Oh well, I'm a probability geek, but not a very good one apparently, haha. I had the methodology mostly right though, just forgot a couple of crucial factors. One was a silly mistake and another one is not as obvious.

 

[Tawnos]

Another reason why I think the Rangers record is much worse than their play.

Right. The Rangers are one game over .500 with the second or third most ROWs in the East and, though I haven't checked, probably the team with the most regulation wins. That they don't push game into OT is definitely concerning, but look at it that way is encouraging.

To OP, what will be really interesting is if the numbers swing back the other way.

 

[iamitter]

Chances of it happening to us:
(27 choose 0)*(.247)^0*(1-.247)^27+(27 choose 1)*(.247)^1*(1-.247)^26=0.004647 (0 or 1 overtime game)
For exactly 1 over time, it's (27 choose 1)*(.247)^1*(1-.247)^26=.004175

The expectation value is quite a bit simpler than you made if we assume a binomial model (number of successes with n independent trials, which we assume here).
It's just E[x]=np or 100*.004647=.4647 expected number of times for this to happen every hundred years for one team.
For every team (assuming 30 teams for 100 years), it'll be just (30/2)*100*.004647=6.97 (how many times it happens to ANY team in 100 years)
or once every 14.3455 years. This has a variance of 6.9381 and standard deviation of 2.63403.

You can expand this a little bit.
You can expect 0 overtime games in the first 27 games for ANY team to occur once every 141.39 years.
Alternatively, you can expect 0, 1 or 2 overtime games in the first 27 games for ANY team to occur once every 2.969 years.


You could use like 3 or 4 different models for this off the top of my head, but there's no good reason not to use the binomial, so that's why I used it. Sorry if there are any math mistakes, I'm in a bit of a rush.


If you're interested, you can also use the negative binomial random variable if you want to find something like the trial number of the r th success (with independence) and probability p.
The formula you'd use then is
P(i)=(i-1 choose r-1)*p^r*(1-p)^(i-r) with i>r. You'd have E[x]=r/p and Var[x]=r*(1-p)/p^2

 

[SnowblindNYR]

Right. The Rangers are one game over .500 with the second or third most ROWs in the East and, though I haven't checked, probably the team with the most regulation wins. That they don't push game into OT is definitely concerning, but look at it that way is encouraging.

To OP, what will be really interesting is if the numbers swing back the other way.

Eh, Tawnos, if you read through the thread you'll notice I overlooked 2 HUGE factors and if you look at ANY 27 game sample across ALL teams and not just 1 team, you get a 99.96% chances that within one 27 game stretch in a given year a team will have 1 or fewer OTs. I forgot to look at all 27 game samples. However, within 1 season there are 56 per team. I in the first post took only 1 27 game stretch and after that took 3 a season. So I got 149 samples to get to over 50% (and that's BARELY 50%). I also forgot to include every team (30) and all 56 samples per team. So that's 1,680 such samples a year. So the chances that a team in 1 27 game sample has only 1 OT team game or fewer is less than 0.5%, but within the season that 1 team will have 1 or fewer OTs in a 27 game stretch becomes (99.95^1,680) which actually is a HUGE number. 99.96%. So the chances that 1 team DOESN'T have any one such 27 game stretch of 1 or fewer OT games is 0.04%. In other words it should pretty much happen every season and the Rangers are nothing special. It just looks weird because it's the first 27 games and since we're only looking at 1 such sample per team (no one remembers 27 game samples in the middle of the season), there's only an approximately 13% chance that a team does it to start the season. So there's a reason it looks weird when we're looking at it with an eye test we're omitting 1,650 samples (1,680-30) because we're only looking at the first 27 games per team.

 

[iamitter]

Eh, Tawnos, if you read through the thread you'll notice I overlooked 2 HUGE factors and if you look at ANY 27 game sample across ALL teams and not just 1 team, you get a 99.96% chances that within one 27 game stretch in a given year a team will have 1 or fewer OTs. I forgot to look at all 27 game samples. However, within 1 season there are 56 per team. I in the first post took only 1 27 game stretch and after that took 3 a season. So I got 149 samples to get to over 50% (and that's BARELY 50%). I also forgot to include every team (30) and all 56 samples per team. So that's 1,680 such samples a year. So the chances that a team in 1 27 game sample has only 1 OT team game or fewer is less than 0.5%, but within the season that 1 team will have 1 or fewer OTs in a 27 game stretch becomes (99.95^1,680) which actually is a HUGE number. 99.96%. So the chances that 1 team DOESN'T have any one such 27 game stretch of 1 or fewer OT games is 0.04%. In other words it should pretty much happen every season and the Rangers are nothing special. It just looks weird because it's the first 27 games and since we're only looking at 1 such sample per team (no one remembers 27 game samples in the middle of the season), there's only an approximately 13% chance that a team does it to start the season. So there's a reason it looks weird when we're looking at it with an eye test we're omitting 1,650 samples (1,680-30) because we're only looking at the first 27 games per team.

I think that's all we really care about

It's kind of like the 50 in 50. Not as memorable if you do it in the middle of the season or across multiple.

 

[SnowblindNYR]

Chances of it happening to us:
(27 choose 0)*(.247)^0*(1-.247)^27+(27 choose 1)*(.247)^1*(1-.247)^26=0.004647 (0 or 1 overtime game)
For exactly 1 over time, it's (27 choose 1)*(.247)^1*(1-.247)^26=.004175

The expectation value is quite a bit simpler than you made if we assume a binomial model (number of successes with n independent trials, which we assume here).
It's just E[x]=np or 100*.004647=.4647 expected number of times for this to happen every hundred years for one team.
For every team (assuming 30 teams for 100 years), it'll be just (30/2)*100*.004647=6.97 (how many times it happens to ANY team in 100 years)
or once every 14.3455 years. This has a variance of 6.9381 and standard deviation of 2.63403.

You can expand this a little bit.
You can expect 0 overtime games in the first 27 games for ANY team to occur once every 141.39 years.
Alternatively, you can expect 0, 1 or 2 overtime games in the first 27 games for ANY team to occur once every 2.969 years.


You could use like 3 or 4 different models for this off the top of my head, but there's no good reason not to use the binomial, so that's why I used it. Sorry if there are any math mistakes, I'm in a bit of a rush.


If you're interested, you can also use the negative binomial random variable if you want to find something like the trial number of the r th success (with independence) and probability p.
The formula you'd use then is
P(i)=(i-1 choose r-1)*p^r*(1-p)^(i-r) with i>r. You'd have E[x]=r/p and Var[x]=r*(1-p)/p^2

Our math seems to be ever so slightly different for 0 or 1. I assume it's a rounding error. I got 0.004659665. Ha, it's funny I completely forgot about binomials. I quite often do probability, but don't use anything other than probability and combinations/permutations. I learned it a while back and it's really simple. I have to look at this a bit more in depth, it's a lot harder to analyze someone's work than do your own (usually).

 

[SnowblindNYR]

I think that's all we really care about

It's kind of like the 50 in 50. Not as memorable if you do it in the middle of the season or across multiple.

Yeah but people think that it's a rarity to go 27 games with only 1 OT game. But there are so many samples in the season, it's not that rare. Funny, how 2 hours ago I had a result that was the opposite of the real answer. Just curious do you work with math or something or are just good at math? I don't really work with math, but it's a hobby of mine. If I did work with math, I'm sure I would have gotten the right answer.

 

[PlamsUnlimited]

To me this says that the Rangers need a boot in their ass to get some exra points. good work

 

[iamitter]

Yeah but people think that it's a rarity to go 27 games with only 1 OT game. But there are so many samples in the season, it's not that rare. Funny, how 2 hours ago I had a result that was the opposite of the real answer. Just curious do you work with math or something or are just good at math? I don't really work with math, but it's a hobby of mine. If I did work with math, I'm sure I would have gotten the right answer.

I'm a senior applied math and physics dual major with a comp sci minor
Sitting in my complex analysis class right now!

 

[SnowblindNYR]

I'm a senior applied math and physics dual major with a comp sci minor
Sitting in my complex analysis class right now!

What do you want to do after college? I'm actually a huge math enthusiast. I have many a times did probability for fun. This stuff is like crack to me. I'm really a numbers guy and want a do over with my career and go into finance (not computational though, just research). I never really got into advanced stats in hockey, I think I may have to.

 

[SnowblindNYR]

Also, how did you get a variance and standard deviation? What sample were you using? I had the list of teams and their OT games and games total, but I didn't share it? How did you get those 2 numbers? I also love the binomial equation, now I can add that to my analysis, thanks!

 

[iamitter]

Also, how did you get a variance and standard deviation? What sample were you using? I had the list of teams and their OT games and games total, but I didn't share it? How did you get those 2 numbers? I also love the binomial equation, now I can add that to my analysis, thanks!

The equation for the variance when using the binomial model is Var(x)=np(1-p) and the standard deviation is just the square root of the variance. I simply used (30/2)*100*27 as my sample size. Each team with 27 games divided by two to account for the dual nature of games, 100 years.

What do you want to do after college? I'm actually a huge math enthusiast. I have many a times did probability for fun. This stuff is like crack to me. I'm really a numbers guy and want a do over with my career and go into finance (not computational though, just research). I never really got into advanced stats in hockey, I think I may have to.

If you would like to learn more about stats I highly recommend the following books:
Probability Theory: The Logic of Science by Jaynes
Naked Statistics: Stripping the Dread from the Data by Wheelan
They are some of the best basic statistics books out there. A calculus background is very useful when you start hitting some of the higher topics.

For data analysis, you absolutely must read
Data Analysis Using SQL and Excel by Linoff
It's a gem.

As for me, I'd like to eventually do advanced sports statistics for a living. I've been working on my own hockey analysis research here at Columbia, which I'll hopefully be able to publish sometime in the middle of next year.

It almost definitely won't work out, though, so I'll likely be looking at consulting/big data (companies like 1010data). In any case, if you really want to get into any analytics, a strong programming background is pretty much a prereq at this point. I'd start off with Python then move on to C++.

 

[SA16]

The equation for the variance when using the binomial model is Var(x)=np(1-p) and the standard deviation is just the square root of the variance. I simply used (30/2)*100*27 as my sample size. Each team with 27 games divided by two to account for the dual nature of games, 100 years.



If you would like to learn more about stats I highly recommend the following books:
Probability Theory: The Logic of Science by Jaynes
Naked Statistics: Stripping the Dread from the Data by Wheelan
They are some of the best basic statistics books out there. A calculus background is very useful when you start hitting some of the higher topics.

For data analysis, you absolutely must read
Data Analysis Using SQL and Excel by Linoff
It's a gem.

As for me, I'd like to eventually do advanced sports statistics for a living. I've been working on my own hockey analysis research here at Columbia, which I'll hopefully be able to publish sometime in the middle of next year.

It almost definitely won't work out, though, so I'll likely be looking at consulting/big data (companies like 1010data). In any case, if you really want to get into any analytics, a strong programming background is pretty much a prereq at this point. I'd start off with Python then move on to C++.

Some years back I was close to going to Columbia until I found out they required Engineers to take foreign language courses
And, as for a sports analytics book, I would recommend The Book by Tom Tango. It gets deep into statistics and makes a lot of conclusions from them. It also shows a lot of the math behind the work. It's solely baseball focused though.