The Data

I downloaded all scores from the 2016 Schaudenfreude season (weeks 1-22, which does not include the playoffs). There are 12 teams, for a total of 264 individual scores.

Distribution of Scores

There are certainly some differences among teams, but with only 22 data points per team, I think the best option is to estimate a single distribution of scores for all teams.

Simulation of the Probability of a Tie - Observed Scores Only

Drawing directly from the empirical distribution of scores (assuming both weeks and teams are exchangeable), I sampled 1,000,000 pairs of scores and calculated the percentage of pairs for which a tie was observed as \(\frac{1}{226}\).

Simulation of the Probability of a Tie - Kernel Density Estimate

Sampling directly from the set of observed scores is not very realistic, because there are scores that were not observed, but are just a likely to occur as the scores that were observed. To correct for that, I computed a kernel density estimate for the score distribution and sampled from that. Since the KDE is continuous, but scores occur in increments of 0.25, I rounded each score to the nearest 0.25. I again sampled 1,000,000 pairs of scores and calculated the percentage of pairs for which a tie was observed, which this time was \(\frac{1}{1739}\) – not too far from my rough estimate of \(\frac{1}{2000}\).