Combinatorics【荐】.pptVIP

Possible poker hands Problem Show how to determine the number of ways in which to get a poker hand containing exactly a pair. Combinatorics If you flip a penny 100 times, how many heads and tales do you expect? Binomial distribution: Independent events: the outcome (H,T) of the second coin does not depend on the outcome of the first. Typical sequence of result of 10 flips: HTTHTTTHTH Given N fair coins, the probability of any given outcome sequence is (1/2)*(1/2)*…*(1/2)=1/2^N The probability of HTTHTTTHTH is (1/2)^10=1/1024 What if order doesn’t matter? Two coins: the possible outcomes are: 1) TT 2) TH 3) HT 4) HH Each with probability ? The probability of one head and one tail is equal to ? since it can happen two different ways. Choosing subsets A set of N elements has 2^N subsets if we include the empty set and the whole set. Think of the set a set of N coins and the “chosen” subset of the ones that will be heads. Binomial coefficients Let X1, X2, X3, ... Xn be a sequence of n independent and identically distributed (i.i.d) random variables each having finite values of expectation μ and variance σ2 0. The central limit theorem states that as the sample size n increases[3] [4] , the distribution of the sample average of these random variables approaches the normal distribution with a mean μ and variance σ2 / n irrespective of the shape of the original distribution. The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work ThéorieAnalytique des Probabilités, which was published in 1812. Laplace expanded De Moivres finding by