Norway-Spain 28-25: How exciting was it?

First published on 21 December 2014.

We all watched Norway-Spain 28-25 today (21-Dec-2014), with Norway winning the European Championship 2014. But how exciting was it? In precisely how dire straits did Norway find herself when Spain was leading 10-5 after 18 minutes, and could we really start relaxing when Norway led 26-22 after 52 minutes?

Foto: Bjørn Langsem

At time \(t\), running from zero to sixty minutes, the two teams have scored respectively \(N(t) \) and $S(t)$ goals. I model these as independent Poisson processes with constant rate $\lambda=27/60$, which is a reasonable assumption regarding matches involving teams that are almost of the same strength. At time $t$ we have thus excitedly observed $N(t)$ and $S(t)$ and frantically speculate about the final scores $$N(60)=N(t)+N' \quad {\rm and} \quad S(60)=S(t)+S', $$ where $N'$ and $S'$ are independent Poisson variates with parameter $\lambda(60-t)$. We may in particular compute 
$$\eqalign{
p_N(t)&=\Pr\{N(t)+N'>S(t)+S'\}
   =\Pr\{N'-S'>S(t)-N(t)\}, \cr
p_S(t)&=\Pr\{N(t)+N'<S(t)+S'\} 
   =\Pr\{N'-S'<S(t)-N(t)\}, \cr
p_E(t)&=\Pr\{N(t)+N'=S(t)+S'\}
   =\Pr\{N'-S'=S(t)-N(t)\}, \cr} $$ in terms of the probability distribution of the Poisson difference $N'-S'$.

I have done so, admittedly a posteriori, just after the match, but these calculations lend themselves easily enough to the construction of a little `app', able to compute, in real time, the winning, losing, draw probabilities in any game, as the goals keep being scored.

For the match at hand we learn that when Spain was leading 10-5 after 18 minutes, the probability that Spain and Norway would win in the end were 76.9% and 18.4%, with 4.7% chance of having a draw. This was the best-looking time point for Spain. When Norway scored to 26-22 at 52 minutes, however, the chances were 90.3% and 4.9% for Norway and Spain for winning (and 4.8% for a draw). Spain came closer, however, and managed to get to 25-26 with only five minutes to go, which translates to winning chances 59.5% and 23.2% for Norway and Spain (and never before in the match had the chance of the match ending in a draw been bigger, at 17.1%).

See Figure B, displaying these three probabilities as a function of match time, and Figure A, which gives the goals scored. I suggest that tv2 makes an app out of this, so that Figures A-B can be displayed, at appropriately precarious moments in the match.

Figure A: Norway (black line) vs. Spain (red line).
Figure B: Real time real excitement plots:  Three probability curves, as function of game time, for Norway winning (black), for Spain winning (red), and for the match ending in a draw (green).

See appendix: http://www.mn.uio.no/math/personer/vit/nils/dokumenter/handball_app.pdf

Tags: Handball, Poisson process, Excitement plot By Nils Lid Hjort
Published Sep. 16, 2015 11:14 AM - Last modified Sep. 16, 2015 12:35 PM
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