CS6015: Linear Algebra and Random Processes

Lecture 23:  Counting Principles : Very Simple Counting, Multiplication Principle

Learning Objectives

Why do we need to learn Counting Principles?

What is the multiplication principle?

Why do we need "Counting"?

 Why do we need Probability Theory?

What is the probability that a statistic computed from a sample is close to that computed from a population?

Population

Sample

Statistics (from sample)

- mean sugar level

- mean no. of runs

- mean agri. yield

- variance in fertility rate 

Compute using Probability Theory

 

 Why do we need Probability Theory?

Machine Learning

 
P(label = cat | image) ?

predict a distribution over classes

 

cat? dog? owl? lion?

 

cat     dog   lion

 

0.7

0.2

0.1

 Why do we need Counting?

What is the probability of getting a heads?

\frac{1}{2}~or~50\%

How did you compute this?

2 possible outcomes: each equally likely

 Why do we need Counting?

What is the probability of getting a 6?

\frac{1}{6}~or~16.67\%

How did you compute this?

6 possible outcomes: each equally likely

 Why do we need Counting?

What is the probability of getting 4 aces?

\frac{1}{n}

But what is         

n is the number of possible outcomes, i.e., all possible combinations of 4 cards

n ?
\cdots

How do you count

n ?

(using principles of counting)

 Why do we need Counting?

Without knowing how to count the number of outcomes we will not be able to compute the probbaility

\cdots

Turns out that there are 270725 ways of selecting 4 cards from 52 cards! (0.00036% chance of getting 4 aces)

Objective of the chapter

Learn how to count the number of outcomes of an experiment

\cdots

Very Simple Counting

Counting: a simple example

How many numbers are there between 73 and 358? (both inclusive)

Easy!

How many numbers are there between 73 and 358 which are divisible by 7 ? (both inclusive)

Hmm, a little hard!

Let's dumb it down even further and start from the absolute basics!

Counting: the simplest example

How many numbers are there between 1 and 358? (both inclusive)

Super Easy! 358

The number of numbers between 1 and n is n (yup, it doesn't get simpler than this)

Counting: a simple example

How many numbers are there between 73 and 358? (both inclusive)

73, 74, 75, ...., ...., ...., 356, 357, 358

We know how to count from 1 to n (can we use that principle here?)

-72

1, 2, 3, ...., ...., ...., 284, 285, 286

Counting: a simple example

How many numbers are there between 73 and 358? (both inclusive)

73, 74, 75, ...., ...., ...., 356, 357, 358

358 - 72 = 358 - (73 - 1) = 358 - 73 + 1 = 286

The number of numbers between k and n is (n-k+1)

How many numbers are there between 73 and 358 which are divisible by 7 ?

77, 84, 91, ...., ...., ...., 343, 350, 357

We know how to count consecutive numbers from k to n (can we use that principle here?)

11, 12, 13, ...., ...., ...., 49, 50, 51

\div 7

(51 - 11 + 1 = 41 numbers)

Counting: a (not so) simple example

Counting: a (not so) simple example

How many numbers are there in this sequence ?

-21, -17, -13, ...., ...., ...., 391, 395, 399

-20, -16, -12, ...., ...., ...., 392, 396, 400

+ 1
\div 4

-5, -4, -3, ...., ...., ...., 98, 99, 100

(100 - (-5) + 1 = 106 numbers)

How many numbers are there in this sequence ?

+ 2
* 12
9\frac{5}{12}, 9\frac{5}{6}, 10\frac{1}{4}, \dots, 21\frac{1}{2}, 21\frac{11}{12}, 22\frac{1}{3}
\frac{113}{12}, \frac{118}{12}, \frac{123}{12}, \dots, \frac{258}{12}, \frac{263}{12}, \frac{268}{12}
113, 118, 123, \dots, 258, 263, 268
115, 120, 125, \dots, 260, 265, 270
\div 5
23, 24, 25, \dots, 52, 53, 54

Counting: a (not so) simple example

The multiplication principle

Can you have a different combo on every day of the month?

South

North

Beverage

Combo:

S

N

B

Then number of ways of making a sequence of independent choices is just the product of the number of choices at each step

How many ways are there of forming such a committee?

8

12

Committee:

B

G

Learning Objectives

Why do we need to learn Counting Principles?

What is the multiplication principle?