4.7.1 Law Of Large Numbers Video
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...