Non-Transitive Dice Challenge
You've probably come here via the dice we handed out at one of our recruiting events. By now you've hopefully noticed what's special about the 4 dice you're looking at: "red beats green beats blue beats white beats red", each with 66.7% probability. They are known as Efron's dice, and you can find some more info on this Wikipedia page.
Your mission, should you choose to accept it, is to generalise the situation. For example, can you improve on the 66.7%? What if the dice are replaced by arbitrary random variables? What is the best "mutual beating probability" if you have only three dice? Or more than four?
Feel free to solve this with pen and paper, or with the help of a computer - we only care about the bottom line. Just tell us what method you use, or provide whatever proof or hand-waving justification you can. Please send your work and your CV to hiring@gresearch.co.uk and we shall be in touch.