New York Rangers statistical oddity (Overtime Games)

[SnowblindNYR]

Hi all,

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.


Edit: It turns out that I forgot two very important factors and was reminded by a couple of fellow fans. I looked at only 1 team and only 1 27 game sample. However, I forgot to include the fact that there are 30 teams and 56 different 27 game samples per team. So in essence there are 1,680 such samples EACH YEAR. Chances that in ANY 27 game stretch in a season by any team, that no team plays only 1 or fewer OTs is an enormous 99.96%. In other words the Rangers aren't special. The probability that we DON'T see at least 1 team go through a 27 game stretch like that in a given season is a paltry 0.04%. Now, the reason it looks improbably to do what the Rangers are doing is because people only look at the first 27 games. The probability that at least 1 team in 30 goes through such a stretch to start a season (first 27 games) is only about 13%.

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[SnowblindNYR]

That's a tad bit misleading. 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]

One more clarification. 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 (roughly). 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 (ever so slightly) in your favor only after 50 years of these 27 game samples.

 

[SnowblindNYR]

Blah, butchered it, overlooked 2 important factors. It's a lot more common than I got.

 

[Bear of Bad News]

Assuming your premises (that I haven't checked, but seem reasonable):

p = 100/405 (approximately 0.247)

Then I get a probability of 0.466% of having zero or one overtime games in a twenty-seven game stretch, or about 1 in 215.

Given 30 NHL teams, the odds that we would see this in a team's first 27 games this year would be about one in seven (assuming independence, which is a slight consideration).

Of course (and where I think that you were going with your comments above), a team has many 27-game stretches over the course of a full season and while those stretches aren't independent of one another (in fact, they overlap by a lot), a team has a better chance of seeing this over the full season (although it's still relatively unlikely for a given team to see it).

 

[Bear of Bad News]

And since I like doing this sort of thing in Excel, here's an estimate of the smallest number of overtime games a team will see, in a 27-game stretch, over the course of an 82-game schedule.

OT Games|Frequency
0|0.68%
1|4.57%
2|13.88%
3|24.06%
4|26.03%
5|18.63%
6|8.67%
7|2.73%
8|0.64%
9|0.09%
10|0.01%

I never saw a minimum stretch of 11 or more in my sample (118,000 seasons).

Stated differently, 0.68% of teams will have at least one 27-game stretch without any overtime games in the course of a season.

4.57% will not have the above, but will have at least one 27-game stretch with exactly one overtime game in the course of a season (what the Rangers have done so far).

13.88% will not have the above, but will have at least one 27-game stretch with exactly two overtime games in the course of a season.

And so forth...

Also note that the estimate I used, a 100/405 probability of an overtime game, is based upon the data you provided (which I haven't verified, but if true, would include the Rangers' season so far - and so the true proportion may be higher).

 

[SnowblindNYR]

I butchered it. Basically, I forgot to include all 30 teams and forgot that there are A LOT of 27 game samples per team. So there are actually 1,680 different 27 game stretches in the NHL (56*30), I basically did 1 team and 1 such stretch. The odds that at least 1 team will see only 1 or fewer OTs in any 27 game stretch is about 99.96%. The reason it looks odd is because people look at only 1 27 game stretch, the first one, no one cares about games 4-30. So chances that a team has 0 or 1 OT games in their first 27 games is in fact only about 13%.

 

[Bear of Bad News]

The odds that at least 1 team will see only 1 or fewer OTs in any 27 game stretch is about 99.96%.

Without seeing your derivation, I think that you're assuming independence here (which is false). See my post two above yours.

 

[SnowblindNYR]

Are you including 56 different 27 game stretches per team multiplied by 30 teams? Our numbers differ, but I feel like we found out similar but different things. I got 99.96% that 1 team will have 0 or 1 OT games. I'm having a hard time getting your numbers. I guess the problem is that 99.96% includes 56 27 game stretches per team.

 

[Bear of Bad News]

I guess the problem is that 99.96% includes 56 27 game stretches per team.

Yes.

And, as I mentioned above, those stretches are nowhere near independent of one another.

For instance, suppose that the first 27-game stretch looks like this (with 1 representing overtime, and 0 representing non-overtime):

100010010110101010100111010

Is the second 27-game stretch (representing games 2 through 28) an independent random sample? No. In fact, you know exactly what 26 of the 27 values are.

You'll find it hard to get an ultimate probability in a non-empirical fashion (although it could be done). I did the work empirically above (as you can see, the answer is significantly different).

 

[SnowblindNYR]

Without seeing your derivation, I think that you're assuming independence here (which is false). See my post two above yours.

Yeah I see what you mean. A team that has 2 OT games in the first 27 games has a higher chance of having 1 or fewer in games 2-28 than a team that has 3 in the first 27 (in fact it'll be 0). But how did you factor in them not being independent? I actually don't think I know how to do that.

 

[Bear of Bad News]

But how did you factor in them not being independent? I actually don't think I know how to do that.

It could be done using probabilistic principles, although with a season of this length, you'd likely need something like Mathematica to express it in a closed form.

I approximated it empirically, using MS-Excel and a pseudo-random number generator.

=RAND() will give you an (essentially) random number uniformly distributed between 0 and 1. From there, it's a simple question of counting up what you want to count.

 

[SnowblindNYR]

I love probability and things like that, but sometimes I bite off way more than I can chew. I feel like I did that there. What do you mean you did it empirically? Did you get actual numbers and do the math from there? Seems like this year 10 OT games (albeit not always in 27 games) was more common than 1. The Kings had 11 and I think another team had 10, maybe not, but the results were skewing upwards.

 

[Bear of Bad News]

"Empirically" means that I simulated the results. I took as given that the probability of an overtime game was 100/405, and I assumed that this probability was the same for each game in the 82-game season.

Do you have Excel? If you type

=IF(RAND()<(100/405),1,0)

into an empty cell, it will return a "1" (meaning an overtime game) or a "0" (meaning a non-overtime game) with the frequencies described above.

If you then put 82 of these next to one another, and (in a different cell) add up those values, that will tell you how many overtime games a team played in that "simulated" season. Summing up 27-game segments (one at a time) gives you something similar to your original question, and then you can play from there.

This is a brief primer on simulating things in Excel:
http://office.microsoft.com/en-us/excel-help/introduction-to-monte-carlo-simulation-HA001111893.aspx

I've never read that before today, but it seems like a great place to poke around and learn some techniques (if you are so inclined).

 

[Bear of Bad News]

This is also a reasonably interesting tutorial:
http://www.aug.edu/~sbajmg/quan6610/Inventory/simulate rev 5 @risk.pdf

I'll look for some sports based examples in Excel when I have available time.

 

[SnowblindNYR]

Thanks. Whenever I try to be ambitious and fail I teeter between feeling silly for being s ambitious and feeling proud for trying to take a risk, but as a pessimist I'm usually closer to feeling silly.

 

[Bear of Bad News]

The only way to learn is to try things outside of your comfort zone.

 

[SnowblindNYR]

I have sat home during weekends and did probability. Some of it I'm sure I did right (deriving the mega millions odds) and some I was wrong, but I think I got right eventually. I once worked an entire Memorial Day on probability (though that was before I really understood most of it, so it may well have been wrong).

 

[SnowblindNYR]

I thought of another problem with this analysis. If one team plays in OT another team has to play in OT with 100% certainty and that's not taken out of the equation.

 

[Bear of Bad News]

True. It probably doesn't affect it much, however.

 

[patnyrnyg]

Really is whacky. 29 games and only 1 OT, no SO's. 15 W's and 14 losses. Detroit has lost more games than they have won, but because almost half their losses were in OT, they have more points. Ridiculous, if you ask me. Devils have won 11 and lost 18, yet they get a participation point for 6 games, they are right behind the Rangers in the standings. Even more ridiculous.

 

[patnyrnyg]

They go 33 games with only 2 going to OT and no shootouts, and then games 34 and 35 both go to a shootout.