As usual, the hydrologic games were held during the HWRS2018 conference dinner, and this time Victoria won. Competitors threw a dart to select an initial loss and rainfall temporal pattern which were used as input to a RORB model to simulate a 1% AEP peak flow. The team with the highest average peak was the winner.

Before the games started, I thought one of the small teams would win. There were only two international competitors and six from Tasmania, but likely over 100 from Victoria. Extreme results are more likely from small samples so I expected small teams to have both the highest and lowest scores with the Victorian team to be in the middle, near the true mean.

There is a nice discussion of the effect of sample size on variation in the wonderful book *Thinking Fast and Slow* by Daniel Kahneman. See the start of Chapter 10 “The law of small numbers”. Also, in Howard Wainer’s article *The Most Dangerous Equation*. Wainer argues that misunderstanding of this effect has caused confusion and wasted effort. For example, the small-school movement was based on the idea that students performed better in smaller compared to larger schools. In the US, grants from the Gates foundation were used to facilitate the conversion of large schools to several smaller schools. However, evidence presented by Wainer shows the small-school effect is likely the result of the extra variation when only a small number of students are assessed. Comparisons of many schools showed the smallest schools achieving both the highest and lowest scores.

Back to the hydrologic games. How likely is it that the mean of a large sample, the Victorians, would be higher than all the means from a number of smaller samples? I don’t know how many people participated in the hydrologic games, so let’s guess. Ben says there were “over 150 players”; I’ve divided these up as shown in Table 1.

Now, to simulate the game we draw random numbers from the normal distribution. The Victorian score will be the mean of 100 random numbers, the NSW score will be mean of 15 random numbers and so on. Victoria wins if its mean exceeds the mean of all the other teams.

Using this approach, the probably of Victoria winning is estimated to be 0.00830 (1 in 120), based on the mean of 100 sets of 100,000 simulations. So its less likely than seeing a 1% AEP flood. If a hydrologist attends 20 hydrologic games during their career, and this game was played every time, the probability that they would see Victoria win at least once is about 15%.

The upshot is that when Victoria won, and provided the games were fair, we witnessed a rare, but not impossible event.

Calculations are available as a gist.