CABT Math 8 - Fundamental Principle of Counting

Some Terms

“Probability is the branch of mathematics that

provide quantitative description of the likely

occurrence of an event.

Outcome – any possible result of an experiment or

operation

Sample space – the complete list of all possible

outcomes of an experiment or operation

Event – refers to any subset of a sample space

Counting – operation used to find the number of

possible outcomes

`

Introduction to Counting and Probability

Counting Problems

Counting problems are of the following kind:

“How many combinations can I make with 5 T-

shirts, 4 pairs of pants, and 3 kinds of shoes?

“How many ways are there to pick starting 5

players out of a 12-player basketball team?”

Most importantly, counting is the basis for

computing probabilities.

Example;:“What is the probability of winning the

lotto?”


Counting Problems

Example:

Ang Carinderia ni Jay ay may breakfast promo

kung saan maaari kang makabuo ng combo meal

mula sa mga sumusunod:

SILOG DRINKS DESSERT

TAPSILOG KAPE SAGING

TOSILOG MILO BROWNIES

LONGSILOG ICED TEA

BANGSILOG


Counting Problems

Example:

Question: If you want to create a combo meal

by choose one of each kind, how many choices

can you have?




LONGSILOG ICED TEA

BANGSILOG


Counting Problems




LONGSILOG ICED TEA

BANGSILOG

TAPSILOG

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES


Counting Problems




LONGSILOG ICED TEA

BANGSILOG

TOSILOG

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES


Counting Problems




LONGSILOG ICED TEA

BANGSILOG

LONGSILOG

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES


Counting Problems




LONGSILOG ICED TEA

BANGSILOG

BANGSILOG

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES


Fundamental Principle of Counting

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES

KAPE

MILO

ICED TEA

SAGING

BROWNIES

SAGING

BROWNIES

SAGING

BROWNIES

TAPSILOG TOSILOG

LONGSILOG BANGSILOG

There are 24 possible combo meals

The Product Rules

The Product Rule:

Suppose that a procedure can be

broken down into two successive

tasks. If there are n1 ways to do the

first task and n2 ways to do the second

task after the first task has been done,

then there are n1n2 ways to do the

procedure.


Generalized product rule:

If we have a procedure consisting of

sequential tasks T1, T2, …, Tn that can

be done in k1, k2, …, kn ways,

respectively, then there are n1 n2 …

nm ways to carry out the procedure.

The Product Rules


Basic Counting Principles

The two product rules are

collectively called the

FUNDAMENTAL PRINCIPLE

OF COUNTING

Woohoo…


Generalized product rule:

If we have a procedure consisting of sequential tasks T1, T2, …, Tm that

can be done in k1, k2, …, kn ways, respectively, then there are n1 n2 …

nm ways to carry out the procedure.

The Product Rules


T1 T2 T3 … TnTasks

k1 k2 k3 … kn

No. of

ways

1 2 3no. of outcomes in all = ... nk k k k

Example 1

If you have 5 T-shirts, 4 pairs of pants,

and 3 pairs of shoes, how many ways

can you choose to wear three of them?


Solution

The number of ways is 5 4 3 60


Example 2

How many outcomes can you

have when you toss:

a. One coin?

b. Two coins?

c. Three coins?


2 (head and tail)

2 x 2 = 4

2 x 2 x 2 = 8


Example 3

How many outcomes can you have

when you toss:

a. One die?

b. Two dice?

c. Three dice?

d. A die and a coin?


6

6 x 6 = 36

6 x 6 x 6 = 216

6 x 2 = 12


Example 4

How many ways can you answer a

a. 20-item true or false quiz?

b. 20-item multiple choice test, with

choices A, B, C, D?



20 x 2 = 40

20 x 4 = 80

Example 5

How many three-digit numbers

can form from the digits 1, 2, 3, 4

if the digits

a. can be repeated?

b. cannot be repeated?



4 x 4 x 4 = 64

4 x 3 x 2 = 24

Example 7

How many three-digit EVEN

numbers can form from the digits

1, 2, 3, 4 if the digits

a. can be repeated?




4 x 4 x 2 = 32

3 x 3 x 2 = 12

Example 8

How many three-digit numbers

can form from the digits 0,1, 2, 3, 4

if the digits

a. can be repeated?



4 x 5 x 5 = 100

4 x 4 x 3 = 48


Check your understanding1. How many subdivision house numbers can

be issued using 1 letter and 3 digits?

2. How many ways can you choose one each

from 10 teachers, 7 staff, and 20 students

to go to an out-of-school meeting?

3. How many 4-digit numbers can be formed

from the digits 1 2, 3, 4, 5 if the digits cannot

be repeated?Answers: 1. 26 x 10 x 10 x 10 = 26,000

2. 10 x 7 x 20 = 1,400

3. 5 x 4 x 3 x 2 = 120


Probability is a relative measure

of expectation or chance that

an event will occur.

Basic Probability


Question: How likely is an event to

occur based on all the possible

outcomes?

Computing Probability

Basic Probability


The probability p than an event can

occur is the ratio of the number of ways

that the event will occur over the

number of possible outcomes S.

number of ways that a certain event will occur

number of possible outcomesp

Example 1

There are two outcomes in a toss of

a coin – head or tail. Thus, the

probability that a head will turn up in

a coin toss is 1 out of 2; that is,


Basic Probability

number of heads 1

number of possible outcomes 2p

Example 2

There are 6 outcomes in a roll of die.

What is the probability of getting a

a. 6?

b. 2 or 3?

c. odd number?

d. 8?


Basic Probability

One out of 6: p = 1/6

Two out of 6: p = 2/6 or 1/3

Three out of 6: p = 3/6 or 1/2

Zero out of 6: p = 0/6 or 0

Letter d is an IMPOSSIBLE EVENT

Example 3A bowl has 5 blue balls, 6 red balls, and 4 green

balls. If you draw a ball at random, what is the

probability that you’ll

a. get a blue ball?

b. get a red ball?

c. a green ball?

d. not get a red ball?

e. get a red or green ball?


Basic Probability

5 out of 15: p = 5/15 = 1/3

6 out of 15: p = 6/15 = 2/5

4 out of 15: p = 4/15

5 + 4 = 9 out of 15: p = 9/15 or 3/5

6 + 4 = 10 out of 15: p = 10/15 = 2/3

Check your understanding1. Two coins are tossed. What is the

probability of getting two tails?

2. In a game of Bingo, what is the probability

that the first ball comes from the letter G?

3. All the three-digit numbers formed by using

the digits 1, 2, 3, and 4 without repeating

digits are put in a bowl. What is the

probability that when a number is drawn, it

is odd?


Check your understanding


1. 1 out of 8: p = 1/8

2. 10 out of 75: p = 10/75 = 2/15

3. Number of 3-digit numbers: 4 x 3 x 2

= 24

Number of odd 3-digit numbers:

3 x 2 x 2 = 12

p = 12/24 = 1/2

CABT Math 8 - Fundamental Principle of Counting

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Transcript of CABT Math 8 - Fundamental Principle of Counting