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 ¼ The probability of one head and one tail is equal to ½ 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 σ 2 > 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 σ 2 / 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 Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician AleksandrLyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.

Galton board illustrated

Second application: card games 5 card poker hands The number ways of choosing 5 cards from a set of 52 cards is “52 choose 5” =2,598,960

Probabilities as proportions Number of favorable outcomes divided by total number of possible outcomes Chance of 4 of a kind: 13 out of 2,598, x10^-6= out of a million

Possible poker hands Straight flush 40 Four of a kind 624 Full house 3,744 Flush (nonconsecutive) 5,108 Straight (mixed) 10,200 Three of a kind 54,912 Two pairs 123,552 One pair 1,098,240 No pairs 1,302,540 Total 2,598,960

How to figure… The number of ways to get a straight… Starting rank: 10 possible A,K,Q,J,10,9,8,7,6,5 Number of ways from a given starting rank: 4x4x4x4x4 = 1024 Total: 10,240 Subtract straight flushes: 10,200

How to figure… The number of ways to get 3 of a kind… Rank: 13 possible Number of a given rank: “4 choose 3” = 4 Number of possibilities of remaining two cards that do not give a pair: 48x44/2 Total: 13x4x48x22=54912

Problem Show how to determine the number of ways in which to get a poker hand containing exactly a pair.