(8.4) the addition priniciple of counting

8/8/2019 (8.4) the Addition Priniciple of Counting

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LESSON (8.4) The

ADDITION PRINCIPLE

OF COUNTING ANDVENN DIAGRAMS



Problem Solving with a Venn

Diagram



Next we will look at Venn Diagrams.

In a Venn Diagram the box represents

the entire sample space.

A B

Members

that fitEvent A

go in this

circle.

Members

that fitEvent B

go in this

circle.



A B A B

Event A and B Event A or B

Which is ³A and B´?

Which is ³A or B´?

This is calledINTERSECTION.

This is calledUNION.



A BA BA BA B

+ _

=

The ADDITION PRINCIPLE OF COUNTING

P(A or B) = P(A) + P(B) - P(A and B)

A B

But we haveadded this piece twice! That isone extra time!

We need tosubtract off the extra

time!



Example #1)

Given the following probabilities:

P(A)=0.8 P(B)=0.3 P(A and B)=0.2

Find the P(A or B).

This can be solved two ways.

1. Using Venn Diagrams2. Using the formula

We will solve it both ways.



Example #1 (continued)

P(A)=0.8 P(B)=0.3 P(A and B)=0.2

Find the P(A or B).

Solution using Venn Diagrams:

A B In this example wewill fill up theVenn Diagram

with probabilities.




A B

First fill inwhere the events

overlap.

The probability that a student fits the event A and B

is 0.2.

That means theentire A circle

must add up to0.8.

0.20.6 0.1

0.1

The probability that a student fits the event B is 0.3.The box represents the entire sample

space and mustadd up to 1.

0.2

0.1

0.10.6

The probability that a student fits the event A is 0.8.

That means theentire B circle

must add up to0.3.


P(A)=0.8 P(B)=0.3 P(A and B)=0.2

Find the P(A or B).



Then find the probability of A or B.

A B

0.20.6 0.10.2

.1

0.10.6

P(A or B) = 0.6 + 0.2 + 0.1

I will start byshading A or B.

Then I will add up the probabilities in

the shaded area.

= 0.9 Answer



Solution using the formula:

P(A or B) = P(A) + P(B) - P(A and B)

= 0.8 + 0.3 - 0.2

= 0.9


P(A)=0.8 P(B)=0.3 P(A and B)=0.2

Find the P(A or B).

Answer



Example #2.)

There are 50 students. 18 are takingEnglish. 23 are taking Math. 10 are

taking English and Math.

If one is selected at random, find theprobability that the student is taking

English or Math.

E = taking English

M = taking Math




E MIn this example

we will fill up theVenn Diagram

with the number of students.

Example #2 (continued) There are 50 students.

18 are taking English. 23 are taking Math. 10

are taking English and Math.

If one is selected at random, find the probabilitythat the student is taking English or Math.




E M

First fill inwhere the events

overlap.

The number of students taking

English and Math

is 10.

That means thenumber of students taking

English must add

up to 18.

108 13

19

The number of students takingMath is 23.

The box represents the entire sample

space and mustadd up to 50.

10

19

138

The number of students takingEnglish is 18.

That means thenumber of

students takingMath must add up

to 23.

Example #2 (continued) There are 50 students.

18 are taking English. 23 are taking Math. 10

are taking English and Math.

If one is selected at random, find the probabilitythat the student is taking English or Math.



Then find the probability of English or Math.

E M

108 1310

19

138

P(E or M) =

I will start byshading E or M.

ThenI

will find the probability in theshaded area.

= 0.62

8 10 13

50



Solution using the formula:

P(E or M) = P(E) + P(M) - P(E and M)

= 0.62

Example #2 (continued) There are 50 students. 18

are taking English. 23 are taking Math. 10 are

taking English and Math.If one is selected at random, find the probability

that the student is taking English or Math.

18 23 10

50 50 50

!



In a class of 30 students, 21 belong to a

sports team,16 belong to the band and4 belong to neither. How many students

belong to both the band and a sports team?



(16)

(30)

(4)(21)

Neither

Both

Whole Class

Sports Band

And that 16 ± 11 = 5 students belong ONLY to the band.This means that 21 ± 11 = 10 students belong ONLY to a sports team.

1110 5

In a class of 30

students, 21

belong to a

sports team,16

belong to the

band and

4 belong to

neither. How

many students

belong to both

the band and a

sports team?

How many students are taking sports and/or band?

30 ± 4 = 26 in the combined circles

How many students in sports AND band?

37 ± 26 = 11 in Both



The end!

(8.4) the addition priniciple of counting

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