Beta Function, Baked Ziti, and British Intelligence (Tangentially)

Today I Learned:
1) The beta function is a really nice function to know when working with binomial distributions. For reference, the beta function is defined as the integral from x = 0 to 1 of [x^a * (1-x)^b] with respect to x. If you look at that thing, it's a heck of a lot like the core of a binomial function, minus the constant-with-respect-to-x combinatorial term.

Conveniently, the Beta function is *also* equal to x!y!/(x + y + 1)!. Taken together, you can turn an integral over the binomial distribution into a relatively straightforward product of factorials.

2) ...a decent vegan alternative to baked ziti is rigatoni stuffed with little blocks of frozen tofu, drowned in tomato sauce with whatever vegetables you like with sauce (for me, it's onions, greek olives, thai chili, and garlic), and bake the whole thing for a while. The tofu stuffing is surprisingly ricotta-like for what it is.

3) When Nazi Germany invaded the Soviet Union during the Second World War, British intelligence estimated that the Soviet Union would be completely taken over in about ten days. US intelligence gave the Soviets a month. This gross misestimation of the Soviet armed forces was based on several factors:

a) During the *last* world war, Russian forces had consistenly underperformed relative to their German counterparts. Early in the war, for instance, a tiny German force was deployed to the Eastern front to stall for time while the actual army in the West beat France (which was supposed to take days or weeks). It ended up crushing several Russian armies and taking serious chunks out of Allied territory in the East.

b) Germany, up to that point in the war, had beaten a number of major nations in ridiculously short amounts of time. They even occupied France -- France! the nation that had ground them to a standstill for *years* just a couple decades before -- in a matter of about a month.

c) Nobody at the time thought that the Soviet Union was a stable enough government to survive a major invasion. It was the product of a revolution from just a couple decades before, had suffered *another* revolution since then, and was not well-loved by many of its member states. A lot of experts thought that the Soviet government would implode in more or less the same way the Czarist regime had near the end of the First World War.