The double minor dilemma

"Jay, I want you to sit by yourself for a while and work through some Poisson distributions for us." - Eric Hartline-USA TODAY Sports

When a double minor gets called at the same time as offsetting penalties, the shorthanded team gets to choose how it gets handled. But is it even really a choice?

Welcome to Outnumbered, my new hub for commentary and analysis. In this first installment, I'll examine a choice that a lot of us didn't even know existed a few days ago.

At 2:48 of the third period on Tuesday, the Panthers vs. Flyers game featured a series of roughing penalties. One was assessed to the Panthers' Tomas Kopecky, and three were given to the Flyers -- two to Jay Rosehill and one to Zac Rinaldo.

It turned out that this scenario gave the Flyers a choice of which of their penalties would be offset by the Kopecky penalty. If it was Rinaldo's, that would leave them down one man for four minutes; if it was one of Rosehill's, that would leave them down two men for two minutes.

The team chose the four-on-five, as everyone would expect. And as it turns out, they were right. But are there any scenarios where the three-on-five would be the right choice?

Let's pull out the analytical tools and see what the numbers say. Or you can skip to the end, if you just want the answer and don't want the math. I won't judge you.

The inputs

To answer this question, we're going to need to know how likely a team is to score in each situation. Using the numbers at Behind the Net, we can work out scoring rates over the last six years:

Goals for per 60 minutes Goals against per 60 minutes
5-on-4 6.14 0.86
5-on-3 20.69 0.27

So at first glance, we'd expect teams to do significantly better with the 5-on-3, since their scoring rate is much more than twice as high. But let's dig deeper and see if there are any surprise plot twists.

First, a note about something not considered here: I'm going to assume that the scoring rates are exactly constant throughout a penalty. That's probably not exactly right -- fatigue likely plays a role in a long power play, after all. It'd be possible to look through historic scoring rates to try to account for this, but I'm not doing that here.

There's another issue for the analysis, one that derives more directly from the rules of the game. Simply using the rate stat doesn't account for the fact that minor penalties end when a goal is scored. That one does matter, and I'll make sure to account for it.

The analysis

Under what scenarios could it conceivably be beneficial to take the 3-on-5?

Given the goal rates above, I'd be surprised if we find that teams are more likely to kill off the 3-on-5. If we're going to find a place where 3-on-5 is preferred, it probably won't be when teams are up by a goal. But maybe getting both penalties to run concurrently reduces the chances of scoring two goals, making 3-on-5 a good choice if you're up by two with four minutes to play?

Let's just see about that.

Because a power play goal ends a minor penalty, a four-minute 5-on-4 is exactly the same as two two-minute penalties. So to work out each possibility, we just need to know the chances of scoring in a 5-on-4.

There's an equation called a Poisson distribution that's perfect for something like this, where we need to turn a continuous rate state (6.14 goals per 60) into a discrete counting stat (did we score a goal or not?). I won't go through that math, but it turns out that teams will go scoreless on a two-minute 5-on-4 about 81.5 percent of the time. So that gives us the following scenarios:

  • 66.4% of the time, they don't score (0.815 * 0.815)
  • 15.1% of the time, they score on the first penalty but not the second (0.185 * 0.815)
  • 15.1% of the time, they score on the second penalty but not the first (0.815 * 0.185)
  • 3.4% of the time, they score twice (0.185 * 0.185)

The 5-on-3 is a little bit trickier. If a team scores 90 seconds in, that leaves the opponent with a 30-second 5-on-4, which makes the arithmetic a little more complicated.

I ran a simulation with a million 5-on-3's to see exactly how things work out. Here's the result:

  • 49.9% of the time, they don't score
  • 44.8% of the time, they score one goal (and still have two minutes of 5-on-5 play to try to score that second goal to tie)
  • 5.3% of the time, they score two goals

So we have our answer: no, a team that's up by two goals should definitely not choose to go 3-on-5 in this scenario. Running both penalties off at once isn't as important as running the clock, and teams kill off more time in the 4-on-5.

What if they're trailing?

If you're behind late in the game, could it be advantageous to choose the 3-on-5 so you get the penalties over with and can get back to trying to score?

Again, we'll assume that the standard goal rates hold. This probably isn't a great assumption -- a team that's leading by a goal with four minutes left will run a cautious power play -- but let's go with it for now.

As before, let's start with the 4-on-5, which is the simpler case. Teams score 0.86 goals per 60 minutes at 4-on-5, and Poisson tells us that gives a 5.6 percent chance of scoring a short-handed goal during the four-minute penalty kill. But then we have to remember that the opponents get four minutes to try to score; taking both teams' scoring into account, we find they have a 3.7 percent chance of outscoring their opponent on a four-minute penalty kill. They might do a bit better than that if they pull their goalie, but it won't help much (and teams often wouldn't do it).

Outscoring the opponent on a two-minute 3-on-5 is even less likely, of course -- it'll happen less than one out of every 200 tries. But choosing the 3-on-5 means you get two minutes of even strength play after the penalties end, and that's where the value comes from.

So our final question is how likely a team is to tie the game up during those last two minutes.

Again, I turned to a simulation. (Just using the Poisson distribution would make things look worse than they are, because it would include nonsensical cases where a team scores a goal to tie it and then later gives up an empty net goal.)

I found that if a team goes with an empty net for the last two minutes, they'll score the tying goal 15.8 percent of the time. As this paper notes, their odds are even better than that when you account for the chance they draw a penalty, but this number will suffice for now.

So now we can put together our 3-on-5 odds with our 6-on-5 odds to figure out how they might tie the game in the last four minutes:

  • Outscore the opponent during the 3-on-5 (0.4%)
  • Kill the penalty, then score with an empty net (49.9% x 15.8% = 7.9%)
  • Give up a goal, then score two with an empty net (44.8% x 1.6% = 0.7%)

The net result is a total of over 9 percent chance of a team that's down by a goal tying the game after electing to play 3-on-5 for two minutes and 6-on-5 for two minutes. Even if they are more too conservative and wait until there is one minute left to pull the goalie, they will still do much better than the 3.7 percent figure we got from playing 4-on-5 for four minutes.

I skipped the math. What's the answer?

It's pretty rare that teams are better off taking the 3-on-5, but it's actually the right move if they're trailing by a goal with just over four minutes left.

It's a lot like pulling the goalie at the end of the game: you're adopting a strategy that makes you more likely to give up a goal, because it also gives you a better chance of scoring, and at that point in the game scoring a goal matters a lot more than giving one up.

Of course, if you're trailing by a goal with just a few minutes left, you probably shouldn't have had Jay Rosehill and Zac Rinaldo gooning it up out there in the first place. But we can get into that some other day.

More from SB Nation NHL:

43-year-old Sergei Federov comes out of retirement

Jonas Hiller's awesome Mighty Ducks mask

Brad Stuart gets 3 games for hit on Rick Nash

Watch Tomas Hertl go between his legs for 4th goal

Will Seth Jones be hockey's first African-American superstar?

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