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PROFESSOR: The law of large numbers gives a precise formal statement of the basic intuitive idea that underlies probability theory, and in particular, our interest in random variables and their expectations-- their means. So let's begin by asking what the mean means. Why are we so interested in it, for example. If you roll a fair die, with faces one through six, the mean value, its expected value is 3 and 1/2. And you'll never roll 3 and 1/2 because there is no 3 and 1/2 face. So why do we care about what this mean is if we're never going to roll it? And the answer is that we believe that after many rolls, if we take the average of the numbers that show on the dice, that average is going to be near the mean. The mean is going to be near 3 and 1/2. Let's look at an even more basic example. If it's a fair die, the probability of rolling a six, as with any other number, is one sixth. And the very meaning of the fact that the probability of rolling a six i...

Newton's third law states that forces always come in equal and opposite pairs. One way we can write that is if you imagine two objects, object one and object two, the force exerted by object one on object two is equal and opposite to the force exerted by object two on object one. This makes explicit that real forces always arise from a physical interaction. For any force in a problem, you should always be able to identify the other member of the interaction pair. Newton's third law is the most subtle and sometimes the most confusing of his three laws of motion so I'd like to do an example that will help clarify how to think about it. So I'm going to pick a very extreme example. Let's imagine the collision between a marble moving this way and a train moving that way. And so we'll say that the mass of the train-- I'll write that as "m sub train"-- and the mass of the marble is m sub marble. And obviously, the mass of the train is much l...