Counting Rules RuleWhen does the rule apply?FormulaExamples Fundamental Counting # of possible compounds from sequences of simple events n 1 *n 2 *…*n k Wardrobes Social Security Numbers License Plates Factorial (!) Counting # of possible arrangements of distinct sequences of outcomes, exhausting all possibilities n! = n*(n-1)*…*2*1 Truffle packaging Permutations ( n P r ) Counting # of possible arrangements of distinct sequences of outcomes, without exhausting all possibilities— selecting “r” items from “n” possibilities—including sequences of the same ingredients in different orders (“order matters”). n P r = n! / (n-r)! Locks Combinations ( n C r ) Counting # of possible arrangements of distinct sequences of outcomes, without exhausting all possibilities— selecting “r” items from “n” possibilities—excluding sequences of the same ingredients in different orders (“order does not matter”). n C r = n! / {(n-r)!*r!} Picnic Lottery

Probability Distributions Describe entire populations X = all items in the probability space P(X) = probabilities are relative frequencies for all outcomes in the probability space 0 ≤ P(X) ≤ 1, for each outcome in the probability space  P(X) = 1, over all outcomes in the probability space Population mean,  =  X  P(X)} Population variance,  2 =  X 2  P(X)} –  2

Example of a discrete probability distribution XP(X)X*P(X)X 2 *P(X)  = Population variance  2 = 7.1 – 4 = 3.1

Binomial Populations Discrete, numerical population Counts of “successful” trials in a mutually exclusive sequence of length “n”. The sequences are made of “n” independent and identical binomial trials. –Binomial trials are categorical simple events –Binomial trials have 2 complement outcomes –Identical trials means that each trial has the same probability, “p”, of a success.

Binomial example A baseball player has a probability of hitting a homerun in each at bat of (p=).08. In a given road trip, this player gets (n=) 15 at bats. homerun NOT homerun.08 = p.92 = 1-p homerun NOT homerun.08 = p.92 = 1-p homerun NOT homerun.08 = p.92 = 1-p … n = 15 XP(X) 0=binomdist(0,15,.08,false) 1=binomdist(1,15,.08,false) 2=binomdist(2,15,.08,false) 3=binomdist(3,15,.08,false) 4=binomdist(4,15,.08,false) 5=binomdist(5,15,.08,false) 6=binomdist(6,15,.08,false) 7=binomdist(7,15,.08,false) 8=binomdist(8,15,.08,false) 9=binomdist(9,15,.08,false) 10=binomdist(10,15,.08,false) 11=binomdist(11,15,.08,false) 12=binomdist(12,15,.08,false) 13=binomdist(13,15,.08,false) 14=binomdist(14,15,.08,false) 15=binomdist(15,15,.08,false)

Binomial populations in excel Binomial probability formula: P(x): “=binomdist(x,n,p,false)” –x = # of successes in n trials –n = # of trials in the binomial sequence –p = probability of a success in a trial –false = logical value to compute marginal, rather than cumulative probability.

Binomial example A baseball player has a probability of hitting a homerun in each at bat of (p=).08. In a given road trip, this player gets (n=) 15 at bats. Question: what is the probability that this ball player hits 2 homeruns in this road trip? Answer: plug in excel the following information … =binomdist(2,15,.08,false) … and you will get …

Binomial parameters Population mean,  :  = n*p Population variance,  2 :  2 = n*p*(1-p)

Examples of parameter computations For the baseball player in the previous example, we expect the player to hit an average of 1.2 (=15*.08)homeruns during his road trip, give or take 1.05 (=square root of 15*.08*.92) homeruns.

Finding binomial probabilities: statcrunch You can also compute binomial probabilities in Statcruch: 1.STAT 2.CALCULATORS 3.BINOMIAL a)SELECT n and p b)SELECT x to be the appropriate binomial count value c)SELECT the appropriate algebraic symbol: =, >, <, ≤, or ≥ 4.COMPUTE Or you can use the excel formula: BINOMDIST(X, N, P, false= or true ≤ )