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PROFESSOR: Certain kinds of random variables keep coming up, so let's look at two basic examples now, namely uniform random variables and binomial random variables. Let's begin with uniform, because we've seen those already. So a uniform random variable means that all the values that it takes, it takes with equal probability. So the threshold variable Z took all the values from 0 to 6 inclusive, each with probability 1/7. So it was a basic example of a uniform variable. And other examples that come up, if D is the outcome of a fair die-- dies are six-sided. Dice are six-sided. So the probability that it comes up 1 or 2 or 6 is 1/6 each. Another game is the four-digit lottery number where it's supposed to be the case that the four digits are each chosen at random, which means that the possibilities range from four 0's up through four 9's for 10,000 numbers. And they're supposed to be all equally likely. So the probability that the lottery winds ...

PROFESSOR: There's other kinds of bogus proofs that come up. Let's just run through this one quickly. Here's a fact that you about, roots of polynomials. Every polynomial has two roots, at least over the complex numbers, over c. And how do you prove that? Well, you just write down the formulas for the roots. You know the quadratic formula. One root is a minus b plus this square root of this quantity of 2a. And the other root is minus b minus this square root of b squared minus 4ac over 2a. And that's the end for the proof. You can just plug-in this formula for r1 for x into this polynomial and it would simplify to be equal to 0, which shows that this is a root. You could plug that one into this formula for x and simplify algebraically and discover it was 0, proving that r2 is a root. We've just proved that every polynomial has two roots. Well, that's not true. We haven't proved it. This is a proof by calculation that has problems. What's th...