"That means that for our posterior from above, if you choose blue, the probability of the majority choosing red is about 0.1817. If you choose red, it's the opposite: the probability that the majority will choose red is 1-0.1817 = 0.8183, over 4 times higher."

It may well be that I don't understand Beta priors, or maybe I just see myself as unique or non-representative. But intuitively, I would not think that my own choice can justify anywhere close to that large of a swing in my estimate of the probabilities of the majority outcome. Maybe I am subconsciously conflating this in my mind with the objective probability that I actually cast the tiebreaking vote, which is very small.

Thank you for making this. I have enjoyed following the D&D.Sci series, even though I don't get around to posting full solutions.

One thing that the series has given me is a better awareness of the limits of data science. Given enough time and effort, you can parse a data set along as many dimensions as you please, but the amount of time and effort needed grows exponentially based on the number of possible variables. If this scenario were a video game, I imagine that just controlling a party through a few different dungeons, and thereby seeing which enemies did high damage against which parties, would quickly give an intuitive sense and at minimum would help a strategist to avoid a lot of clear mistakes. The same goes for the League of Defenders of the Storm scenario -- an actual player of the game would quickly learn that level 1's beat their corresponding level 6's in play, whereas that fact wasn't as obvious from observing the overall data set.

So both on-the-ground experience and data science have their uses. It's valuable to practice what can be done with data science alone, but one key takeaway from this series is that if I'm ever making a bet with my own life ( (or just a lot of money) on the line, I pray I'll get a chance to practice my strategy as well as to observe the relevant data.