Why ask?

It’s 2–2 late in the third period, and the broadcaster says the home team is “due” for a goal. But if goals are truly random, there is no such thing as “due.” The puck has no memory.

How statistics can help

• Model: treat goals as random events using the Poisson distribution
• Estimate: find the average scoring rate from real games
• Test: check the model against all 1,312 games of the 2025–26 season
• Predict: use it to forecast scores and overtime games, then compare with what really happened

The Poisson distribution models counts of rare events in a fixed window of time, such as calls to a help desk, typos on a page, or goals in a hockey game.

It assumes events:

• happen one at a time
• arrive at a constant average rate \(\lambda\)
• are independent of each other

Hockey looks like perfect Poisson territory. A game has 3,600 seconds and each team takes about 30 shots, but only about 1 in 10 goes in. That means lots of chances, each with a tiny probability.

Slice the game into \(n\) tiny moments, each with a goal probability of \(\lambda / n\). The number of goals is Binomial, and as the slices shrink:

\[P(X = k) = \binom{n}{k}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k} \;\xrightarrow{\;n \to \infty\;}\; \frac{e^{-\lambda}\lambda^k}{k!}, \qquad k = 0, 1, 2, \dots\]

The Poisson’s signature property:

\[E[X] = \lambda \qquad \text{and} \qquad \operatorname{Var}(X) = \lambda\]

• 1,312 games from the 2025–26 NHL regular season (Hockey-Reference.com)
• Each game gives two counts (home and away), for a total of 2,624 team-games
• We count regulation goals only, so every game is the same 60-minute window

The shootout problem: 207 games ended in overtime and 119 in a shootout. The winner gets +1 on the final score, but a shootout “goal” is not scored during play.

For goal counts \(x_1, \dots, x_n\), the log-likelihood is

\[\ell(\lambda) = \sum_{i=1}^{n}\left(x_i \log\lambda - \lambda - \log x_i!\right) \quad\Rightarrow\quad \ell'(\lambda) = \frac{\sum x_i}{\lambda} - n = 0 \quad\Rightarrow\quad \hat\lambda = \bar{x}\]

The best estimate of the scoring rate is just the average goals per game:

\[\hat\lambda = 3.003, \qquad \text{SE}(\hat\lambda) = \sqrt{\hat\lambda / n} = 0.034, \qquad 95\%\ \text{CI: } (2.94,\ 3.07)\]

3 goals was the most common result: 606 team-games, compared with 588 predicted by the model. In every group, the model lands within about 35 games of reality, out of 2,624 team-games. The biggest misses: teams scored exactly 1 goal more often than predicted (425 vs. 391), and exactly 4 goals less often (406 vs. 441).

\(H_0\): goals per team follow a Poisson(\(\lambda\)) distribution.

\[X^2 = \sum_{k} \frac{(O_k - E_k)^2}{E_k} = 12.03, \qquad df = 8 \text{ groups} - 1 - 1 \text{ estimated } \lambda = 6, \qquad p = 0.061\]

Home teams scored 3.07 regulation goals per game and away teams scored 2.94, a gap of 0.13 goals. Over a team’s 41 home games, that adds up to about 5.5 extra goals per season. The 95% CI for the gap, (0.002, 0.268), stays just above 0, so home ice is a real but small advantage.

Surface = Poisson model, gold dots = real games. Drag to rotate and hover for details. Notice the tied scores along the diagonal: many of those dots float above the surface.

Most common score after 60 minutes: 3–3 (9% of games).

If home and away goals were independent Poissons, the chance of a tie after 60 minutes is

\[P(\text{tied after 60}) = \sum_{k=0}^{\infty} P(H = k)\,P(A = k)\]

The model expected about 218 games to go past regulation. In reality, 326 did.

• The gap: 326 of 1,312 games were tied after 60 minutes (207 OT + 119 shootouts). That is about 108 more than the model’s 218, or roughly 1 extra overtime game for every 12 games played.
• Where they came from: the model over-predicted regulation wins. Home leads dropped from 43.8% predicted to 40%, and away leads dropped from 39.5% to 35.1%.
• Why: the model assumes the two teams’ goals are independent, but real teams react to the score:
  • In a tie late in the game, both teams play safe, because getting to overtime guarantees a point.
  • A team that is behind pulls its goalie and pushes hard to tie it.

• Goals per team are close to Poisson. Mean = 3, variance = 2.96 (ratio 0.99), and the chi-square test did not reject the model (p = 0.061).
• A typical team scores about 3 goals in regulation, 95% CI (2.94, 3.07).
• Home ice is worth about 0.13 goals per game, roughly 5.5 extra goals per season.
• The model misses ties. It predicted 16.6% of games would go to overtime, but 24.8% did.

Data: Hockey-Reference.com, 2025–26 NHL regular season. Built with R, ggplot2, and plotly.

• Poisson fits for the last 3 seasons: variance ≈ mean (0.99–1.03), and no season rejects the model (p = 0.234, 0.055, 0.061).
• Home ice always helps: 0.21, 0.30, then 0.13 goals per game.
• Ties always beat the model: about 4 points above it in the first two seasons, and 8 points above it in 2025–26, the stickiest season of the three.