## The Probability That Plank or Taylor Make an Accurate Prediction

At the end of each gameday Plank or Taylor predict the the final score and the goal scorers. A question came up in the comments section a few days ago about the probability that Plank or Taylor's prediction comes true. Although we all know that Plank and Taylor have supra-human hockey insight (Taylor's powers are so mighty that he has to predict poorly, lest the NHL catch on), we can still test if they have been using their hockey super powers responsibly.

This also gives us an opportunity to talk about Poisson distributions and their relation to hockey goals. The Poisson is used exclusively for random discrete events that occur in small frequencies. There is a set of criteria that must be met in order for a distribution to fall into a Poisson curve, most notably the events must be descrete and independent from each other, (eg. 2 goals cannot be scored at the same time), I'll defer you to wikipedia for more information regarding the distribution.

Can we predict the number of goals scored using the Poisson distribution? or better yet, does the number of goals scored by a team fit the Poisson Distribution? I'm certainly not the first person to ask this question, but instead of blindly accepting the answer I decided to exam it for myself. I took data from the 2007-2010 regular seasons,and determined the number of goals scored by each team in individual games. (N apprx 10,000) I then calculated the mean, generated a histogram of that data, and compared it to the Poisson Distribution with a mean equal to the league average. The results are shown below.

### 2007-2010 Goals/Game for home and away teams

Goals scored Frequency Cumulative % Cumulative Poisson%
0 534 5.56% 4.67%
1 1334 19.47% 18.99%
2 1900 39.27% 40.91%
3 2192 62.11% 63.30%
4 1702 79.85% 80.45%
5 1108 91.39% 90.96%
6 554 97.17% 96.32%
7 190 99.15% 98.67%
8 60 99.77% 99.57%
9 16 99.94% 99.87%
More 6 100.00%

We can see that the Poisson Distribution fits the curve of hockey goals relatively well. Now that we've established that; we can use the Poisson Distribution to determine the probability that a team will score x goals. Each team has their own mean (average) number of goals scored per game, and therefore each team has their own Poisson distribution. In order to get an accurate prediction though, we have to adjust each team's goals/game to account for randomness, and the other team's defensive ability. We previously looked at reliability for goal scoring, and noted that we need to regress goals/game to league average by some "x" percent based on the number of games that have been played. After all this we can plug in Plank or Taylor's prediction into each team's regressed Poisson distribution to determine the probability of that final score coming true. We can use the same principles to evaluate goal scorer predictions, and arrive at a final overall probability.

A working example from Game 1 of the series, wherein Plank predicted a 3-1 win.

### Probability of Predicting goals scored for each team

Goals Scored SJ Prob STL Prob
0 0.102 0.081
1 0.233 0.204
2 0.266 0.256
3 0.202 0.214
4 0.115 0.134
5 0.052 0.067
6 0.020 0.028
7 0.006 0.010
8 0.002 0.003
9 0.000 0.001
Regressed Mean 2.28 2.51

### Probability of a Player Scoring 1 or More Goals

Goal Pavs Couture Braun
0 0.685 0.679 0.970
1 0.259 0.263 0.029
2 0.049 0.051 0.000
3 0.006 0.007 0.000
4 0.001 0.001 0.000
5 0.000 0.000 0.000

We generate the probability that each team scores a certain number of goals, listed in the tables above, then multiply those probabilities together to find the probability of accurately predicting the final score. We can then determine the probability that Pavelski, Braun,and Couture score from thier Poisson distributions. We multiply these all together to determine the final probability.

Plank's prediction for Game 1 was a 3-1 win by the sharks. The probability of a 3-1 victory was 0.20 x 0.204 = 0.041or 4.1%. The probability that Pavelski (0.26) and Couture (0.26) and Braun (0.03) score was 0.26 x 0.26 x 0.03 = 0.00203 or 2.03 x 10^-3. Taking it all together we arrive with the probability of an accurate prediction somewhere around 8.31 x 10^-5.

So, we examined the Poisson distribution, its relation to hockey scoring, and used that to determine the probability that Plank and Taylor make an accurate prediction.. Based on the calculated probabilities we can infer that a prediction will be accurate in about 1 in every 20 games, give or take, and if you include goal scorers, far, far less, more like 1 in 10,000, if chosen by random. If Plank or Taylor predict with greater frequency, we must conclude they have can see into the future, and have supra-human hockey knowledge.

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