counting rules rulewhen does the rule apply?formulaexamples fundamental counting # of possible...

Counting Rules

Rule When does the rule apply? Formula Examples

FundamentalCounting # of possible compounds from sequences of simple events

n1*n2*…*nk

•Wardrobes•Social Security Numbers

•License Plates

Factorial (!)Counting # of possible arrangements of distinct sequences of outcomes,

exhausting all possibilitiesn! = n*(n-1)*…*2*1 •Truffle packaging

Permutations ( nPr)

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”).

nPr = n! / (n-r)! •Locks

Combinations (nCr)

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”).

nCr = 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, = XP(X)}• Population variance, 2 = X2P(X)} – 2

Example of a discrete probability distribution

X P(X) X*P(X) X2*P(X)

0 .2 0 0

1 .3 .3 .3

2 .2 .4 .8

3 .1 .3 .9

4 .05 .2 .8

5 .1 .5 2.5

6 .05 .3 1.8

= 2 7.1

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

X P(X)

0 =binomdist(0,15,.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 …. 0.227306

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. BINOMIALa) 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≤)

counting rules rulewhen does the rule apply?formulaexamples fundamental counting # of possible...

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