Borel-Cantelli Lemma & Law of Large Numbers

This lecture introduces the Borel-Cantelli lemma, which relates convergence in probability to almost sure convergence. The lemma states that if the sum of probabilities of a sequence of events is finite, then the probability of the set of outcomes that belong to an infinite number of events is zero. The instructor explains the concept using a visual analogy with discs in a set. The lecture then delves into the two laws of large numbers: the weak law and the strong law. The weak law states that the empirical average of a sequence of random variables converges in probability to the expected value of the first random variable, while the strong law asserts almost sure convergence. The differences between the two laws and their historical significance are also discussed.