This problem derives an intuitive law of probability known as the law of large numbers from Chebyshev's law. Informally, the law of large numbers says if you repeat an experiment many times, the fraction of the time that an event occurs is very likely to be close to the probability of the event. In particular, we shall prove that for any positive number s, no matter how small, by making the number n independent trials in a sequence of independent trials large enough, we can make the probability that the number X of successes is between np - ns and np + ns as close to 1 as we choose. For example, we can make the probability that the number of successes is within 1% (or 0.1 per cent) of the expected number as close to 1 as we wish.
(a) Show that the probability that |X(x) - np| ≥ sn is no more than p(1 - p)/s2n.
(b) Explain why this means that we can make the probability that X(x) is between np-sn and np + sn as close to 1 as we want by making n large.