I'm pretty sure that page 12 math is correct for the lottery I know. Translator group is seem to apply it for another kind of lottery, which I don't know, and can't derived from the math. Something like diminishing rewards after each round, I think.
On the other hand, math on page 13 by translators give a better view for bigger picture by incorporating cost into equation and results in expecting value of the whole transaction, while the author version is just expected value for the rewards alone. But both are the same thing nonetheless.
Lancaster equation for melee fight isn't something news at all, as it could be easily intuitively derived. It's also rely on unrealistic assumptions that there are no other factors in battle, and even not counting ranged weapons. It is unreasonable to think that princess isn't already aware of it.
I hope he introduced Lancaster equations for ranged exchange instead.
I really hope that the author wouldn't write this by just nerf intelligence of medieval people. Granted they are declined from roman era but still assuming they cannot estimate numerical advantages or expecting value is probably too far. There is still a lot of math advances easily applicable in that era. Logic and game theory of course the first candidates, but mc clearly seems capable of binary calculations and thus, analog computing. With better calculating power, he can initiate calculus revolution, which will put the nation far, far ahead of neighbours. Just by a classic pandemic arc with calculated R value and SIR model, things will be a lot different.