When to pull the goalie? Solved.

[Steady Eddy]

Any hockey fans out there? Math nerds? Both? Great!

Consider a team who's average score against an opponent is 3-3. That's a total of 6 goals per 60 minutes, or .1 goal per minute. The probability that a goal will be scored in "t" minutes is given by (1 - e^-.1t)

Since our team scores half of the goals, the probability that our team will score the next goal is .5(1 - e^-.1t)

If you're one goal behind and there's plenty of time, the probability of tieing the score (maybe only a temporary tie) is close to .5, but as time starts running out, your probability of doing this gets smaller.

Suppose you know that with your goalie out of the game, and an extra attacker replacing him you'd lose 15 - 45 on average. That's a total of 60 goals in 60 minutes, or 1 goal per minute. The probability that a goal will be scored in "t" minutes is (1 - e^-t). But since now our team is only scoring on fourth of the goals, the chance that we score the next goal is .25(1 - e^-t). If a lot of time is left, our chance of tieing the game is only .25, but if time is short, this expression will give higher values than the other.

To find the optimal time to yank the goalie, let y = .25(1 - e^-t) - .5(1 - e^-.1t), find y', and set it equal to zero. You'll get t = ln(5)/.9, or about 1.7 minutes.

BTW, this doesn't depend on the parameters I used. They are only for illustration. They can be replaced with variables and you can still derive the optimal time in the manner I demonstrated.

BTW II, I've looked on the net and haven't seen a formula. I've seen empirical data, instead. But that data doesn't contradict this formula.

 

[stringertom]

In hockey, the math is not that complicated...down a goal with a minute left, you hope to clear the puck out of your zone, throw it in deep and have the goalie skate like hell for the bench. Then it's six-on-five for sixty or so frantic seconds...a diluted power play (vs 5-on-4) but with one big advantage of icing being waived off.

 

[Steady Eddy]

But why sixty seconds? Maybe 120 seconds would be even better? And if you tie it up, you can put your goalie back in.

Maybe he should be benched w/ 5 minutes to go? After all, it doesn't matter if you lose by 1 or 2, a loss is a loss.

 

[stringertom]

It's about how much confidence the coach has in the conditioning of his top offensive line...you rarely get a stoppage to shuffle a fresh line and legs go after more than a minute shift. Ninety seconds? Yeah, but that's when the gas tank empties, especially at the end of a game.

 

[Steady Eddy]

I hadn't thought of that. So that tactic can't really go on for much more than a minute?

 

[stringertom]

I was privileged to get an inside look at the NY Islanders franchise in their dynastic era in the early 80's. I watched many a home game from the Zamboni ramp at one of the goal lines and right behind the boards. Seeing these guys that close makes it easy to be aware of their threshhold of endurance. They literally are blue in the face while heading for the bench after a grinding shift, sometimes not even 90 seconds long. They built their success not only on HOF-level talent but four-line depth so Coach Arbour could shuffle the deck more often and keep fresh legs out there. The irony...in between periods one of their HOFers could be seen in the locker room chain smoking Marlboro Reds. Somehow, I don't think that's happening these days!

 

[hollywood9826]

I always thought of pulling goalie as a similar desperation tactic to throwing a hail mary in pro football. Its last resort effort and therfore should only be depolyed when there is seemingly no other options.

At the 5 minute mark the regular guys still have time to score the goal and stuff can happen still. With 1 minute left all otheroptions are prety much out the window so you throw caution to the winmd and go for it.

 

[Bdarb]

Like someone said, your first line realistically can put in a 60 second high end shift with about 30-45 seconds thereafter. If you pull your keeper too soon, you inevitably will be scored on unless you do so first and within the minute (maybe one and a half) because realistically, no team, a man up or not is going to hang on to the puck relatively uncontested for 2 minutes straight, Nevermind five! If they did have the chance to change, they would most likely have to chance giving up possession of the puck which at this point in the game is essentially a loss.

I like the approach you took to this, but as is the case in most sports most of the time, some level of human rationale must be factored in. It looks great theoretically but common sense dictates that this just couldn't work. I feel like this kind of formula must have been derived a while back, especially because it's interesting that as you said it comes out with about 1.7 minutes being the optimum time which in my experience is right on the money.

Great thread! Not enough hockey on these boards.. do you still play?

 

[max]

I used to play adult hockey at the local rink: listen to stringertom.

It's a "Hail Mary" kind of play.

Talk about stamina! Just recall the last Stanley Cup series! Whew!

 

[Steady Eddy]

I never played organized hockey. Just pick up games at the rink behind the school.

Remember, you don't have to keep your goalie out of the rest of the game. If you score next and tie it, then put him back in. If you get scored on, I think it's over. A two goal margin is pretty hard to eliminate late in a game.

 

[Bdarb]

I never played organized hockey. Just pick up games at the rink behind the school.

Remember, you don't have to keep your goalie out of the rest of the game. If you score next and tie it, then put him back in. If you get scored on, I think it's over. A two goal margin is pretty hard to eliminate late in a game.

not as hard as you might think. Ask alot of hockey players, 3-1 is the worst lead in hockey because it can flip in a second. The bruins had this several times last year where they scored multiple empty net goals and I've seen it happen more times than I can remember while playing. It's just such a fast paced game.

And yes, I know he can be put back in haha.


EDIT: just saw you're in Arizona, Coyotes are my team!

 

[stringertom]

3-1 Isles was the score entering Period 3 of the Cup-deciding game vs Philly in '80. Two minutes later it was tied, without pulling the Michelin man. Crazy next 25 minutes of hockey and it was worth it being there when Mr. OT (Bobby Nystrom) beat Peeters cleanly...the loudest noise humans have ever made. Total BEDLAM!

 

[Bdarb]

3-1 Isles was the score entering Period 3 of the Cup-deciding game vs Philly in '80. Two minutes later it was tied, without pulling the Michelin man. Crazy next 25 minutes of hockey and it was worth it being there when Mr. OT (Bobby Nystrom) beat Peeters cleanly...the loudest noise humans have ever made. Total BEDLAM!

brought home the first of four consecutive cups for the Islanders!! His son Eric Nystrom (flames) is an animal too.


Bobby was also one of the last players to not wear a helmet for all you useless info fans

 

[Steady Eddy]

^^Ah the old days. I remember watching the Minnesota North Stars back in 1970, and only one player was wearing a helmet. One of our goalies, Gump Worsley, didn't even wear a face mask!

 

[r2473]

http://www.urbandictionary.com/define.php?term=pull+the+goalie

 

[stringertom]

^^Ah the old days. I remember watching the Minnesota North Stars back in 1970, and only one player was wearing a helmet. One of our goalies, Gump Worsley, didn't even wear a face mask!

Bruins' Cheevers (IIRC) used to decorate his mask with stitch marks for every puck he took on the face.

 

[Bdarb]

I'm on my phone, so ill post the link in a bit, but please anyone look up "Clark the Canadian hockey goalie" on YouTube. You will NOT be disappointed

 

[maverick66]

Talk about stamina! Just recall the last Stanley Cup series! Whew!

The injuries that Patrice Bergeron had going into it. No idea how you play at that level with a freaking hole in your lung.

http://www.urbandictionary.com/define.php?term=pull+the+goalie

I am pretty proud that I have never once heard the term used in any of those ways. WTF is wrong with this world.

 

[stringertom]

Just gave a pedicab ride to an ex-NHLer who stated Mr. OT Nystrom broke his nose in a five-for-fighting skirmish. I meet some colorful characters out here. Two months ago, I drove a LI businessman who pulled up Mike Bossy's number on his iPhone log...plays golf with him regularly and, yeah, Bossy is still chain smoking his Reds!

 

[r2473]

I am pretty proud that I have never once heard the term used in any of those ways. WTF is wrong with this world.

You can pretty much type any word/phrase into urban dictionary and get some result. Nobody uses these terms in these ways.