## NHL In-Game Win Probability

I was at an NHL game the other night, and with the score 2-0 someone asked me, “So Mr. Win Probability, what’s the chance the Capitals win?” I was caught off guard, and after I choked out, “I…don’t…know…,” I experienced the horror that is not knowing the exact up-to-the-second win probability of a sporting contest. Don’t let this happen to you.

The anxiety and shame lasted for two days straight. I kept blaming myself and replaying the incident over and over in my head. The only way to cure my depression was to build a win probability model for NHL hockey.

Unlike my previous models for basketball and football which were empirically based, my hockey model is theoretical. In other words, instead of being based on a massive database of actual previous games, the probabilities are calculated based on a Poisson scoring distribution. The distribution is calculated using the average goals scored per minute in the 2008-9 NHL season. It’s an extension of the model I developed in this post.

Teams score an average of 2.79 goals per 60 minutes of regulation time, which is equal to 0.0465 goals per minute. A Poisson distribution based on that per-minute scoring rate and the time remaining in the game yields the probabilities of each team scoring each number of possible goals by the end of the game. Summing up all the probabilities of all the possible combinations of final scores gives the game’s win probability.

Here’s the graph:

There are a couple wrinkles to address. First, there are power plays. When a team as a man advantage on the ice, it’s much more likely to score. About one in five power plays results in a goal for the team with the advantage. Only about 2% of the time the short-handed team will score. So at the start of a power play, a rough approximation would put the win probability a little less than one fifth of the way toward the next best curve.

For example, if the score is 2-0 with 30 minutes remaining in the game, the win probability would normally be about 13% for the trailing team (the red line). But at the beginning of a power play, the trailing team’s win probability would jump about a fifth of the way up to the ‘down by 1’ line (blue). A rough approximation puts the new win probability at 16%. Then as the power play expires and there’s no score, the win probability would gradually return to the ‘down by 2’ line.

Second, there is the ‘end-game,’ when teams down by a goal will pull their goalie in favor of an additional skater. That would increase the win probability of the trailing team slightly, but only half as much as you might expect. They’d still only be buying an opportunity in overtime. But it could still be factored in. Before I do, I’d need some data on end-game goals.

One advantage of a theoretical approach over an empirical model is that team strength can be factored in far more easily. In an empirical model, when you divide up the data by various classes of team strength, the data is sliced into tiny fragments, usually with very small and unreliable sample sizes. Theoretical formula-based models don’t suffer from that problem. I can simply adjust the mean goals scored and goals allowed for any particular opponent, then rerun the model. The resulting model would be tailored to the specific match-up instead of a generic model for the league as a whole. Home ice advantage can be factored in with a similar approach.

Remember, WPD (Win Probability Dysfunction) can happen at any time, and it’s nothing to be ashamed of. Don't analyze win probability graphs if you take nitrates, often prescribed for chest pain, as this may cause a sudden, unsafe drop in blood pressure. Discuss your health with your doctor to ensure that you are healthy enough to view win probability graphs. If you experience chest pain, nausea, or any other discomforts during a sporting contest, seek immediate medical help. In the rare event of viewing win probability graphs more than 4 hours, seek immediate medical help to avoid long-term injury.

Live NHL win probability graphs now online.

### 6 Responses to “NHL In-Game Win Probability”

1. Anonymous says:

what were the chances the devils would lose that game (and the series) a goal up with 1:30 to play?

2. Zach says:

It looks as if New Jersey had a 97-98% chance of winning up one with 1:30 left...

Brian, I assume the probability is simply 50% when tied?

3. Anonymous says:

Wouldn't the Poisson distribution and goals scored per minute need to be based upon the average of each individual team, rather than the NHL average?

4. Brian Burke says:

For now there's no home ice or team strength factored in. So yes, any tied game will be 50/50.

5. Mark M says:

Regarding factoring in team strength. One big problem I've noticed looking at football (soccer) numbers, I think this is something you've touched on before Brian with home-team advantage, is that the further into a game you get, the less 'team strength' (as defined by previous games) affects the win probability.

I'm not sure how easy the drop-off effect would be to analyse, or indeed whether it's worth looking at (most interest is in the end-game WP and those are pretty much 100% dictated by the match situation).

6. Anonymous says: