Normal Density Curve

68 % 95 % 99.7 %

Normal Approximation to the Binomial Distribution For n independent trials with success probability p where m = np is the mean and s = (npq)1/2 is the standard deviation.

Law of Large Numbers Informal: If n is large, the proportion of successes in n Bernoulli trials will be very close to p. Formal: For Bernoulli trials with n and p, as n  , for all  > 0, where k is the number of successes in the n trials.

Normal Approximation What happens to the binomial distribution when p is very small or large (i.e., close to 0 or 1)?

Poisson Approximation to the Binomial Distribution If n is large and p is small, the distribution of the number of successes, k, in n Bernoulli trials is largely determined by the mean m = np: In general, the Poisson approximation to the binomial will be excellent when n  100 and m  10.

Exercise 3000 people are watching a parade on a hot summer day. Let’s assume the probability that any one of the 3000 persons watching the parade will collapse from heat exhaustion is 0.005, and that people collapse independent of one another. What’s the probability that 4 people collapse?