stat 515 fa 2021 lec 02 slides counting rules

STAT 515 fa 2021 Lec 02 slides

Counting rules

Karl B. Gregory

University of South Carolina

These slides are an instructional aid; their sole purpose is to display, during the lecture,definitions, plots, results, etc. which take too much time to write by hand on the blackboard.

They are not intended to explain or expound on any material.

Karl B. Gregory (U. of South Carolina) STAT 515 fa 2021 Lec 02 slides 1 / 8

Counting rules

Motivation to study counting rulesIf all outcomes in S are equally likely, for any event A ⇢ S , we have

P(A) =#{sample points in A}#{sample points in S} .

This leads to an interest in counting rules.


Counting rules

Fundamental theorem of countingIf a job consists of K tasks such that the tasks may be completed in n1, . . . , nKways, respectively, then there are

KY

k=1

nk = n1 ⇥ n2 ⇥ · · ·⇥ nK

ways to do the job.

Exercise: You are confronted with the following sequence of choices:

1 Barbacoa, chicken, carnitas, or veggies

2 White or brown rice

3 Black or pinto beans

4 Spicy, medium, or mild

5 To pay extra for guacamole or not to pay extra for guacamole

In which restaurant are you ordering food?


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Counting rules

PermutationDraw r elements of N elements without replacement and arrange them in some

order. The # ways is

N(N � 1) · · · (N � r + 1) =N!

(N � r)!.

Exercise: You must compose a ballad in the key of G, i.e. from the chords

G Am Bm C D Em F#dim

1 In how many ways can four distinct chords be chosen to begin your song?

2 What if you can’t play F#dim?


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Counting rules

Exercise: Jane Austen wrote the following novels:

Pride and Prejudice

Emma

Sense and Sensibility

Persuasion

Northanger Abbey

Mansfield Park

1 In how many ways can you read them all this semester?

2 In how many ways can you read them all such that you read Pride and

Prejudice, Emma, and Sense and Sensibility without reading any of the others

in between?


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Counting rules

PartitionPlace N balls into K urns such that the K urns receive n1, . . . , nK balls, where

N = n1 + · · ·+ nN . The # ways is

N!

n1!n2! · · · nK !.

Exercise: Suppose 12 people are randomly assigned to ride in 3 vehicles taking 4,

5, and 3 passengers, respectively.

1 In how many ways can the passengers be assigned to the different vehicles?

2 If you and your friend are among these people, with what probability will the

two of you ride in the same vehicle?


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Counting rules

CombinationDraw r elements from N without replacement and without regard to their order.

The # ways is ✓N

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N!

r !(N � r)!.

This is just a partition with only 2 “urns”.

Excercise: For German class you must watch 5 episodes from among 10 available

episodes of Bares fur Rares, 8 of Betty’s Diagnose, and 12 of Haustier Check.

1 In how many ways can you choose

I 5 episodes of Bares fur Rares?

I 3 episodes of Haustier Check and 2 episodes of Betty’s Diagnose?

2 If you choose 5 episodes at random, with what probability do you

I not watch any episodes of Haustier Check?

I watch at least one episode of Bares fur Rares?

I binge entirely on Betty’s Diagnose?


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Counting rules

Exercise: If dealt 5 cards from a 52-card deck, what is the probability of getting

1 the ace of diamonds?

2 at least one ace?


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stat 515 fa 2021 lec 02 slides counting rules

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