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This pdf covers the following topics:-

Complementary Events

Addition Rule

Disjoint Events

Non-Disjoint Events

Complementary Events

Addition Rule

Disjoint Events

Non-Disjoint Events

1.
Complementary

Events

Addition rule

Events

Addition rule

2.
Complementary Events

the events of one outcome happening

and that outcomes not happening

are complementary (opposite)

( not E is contrary event to E )

E Not E

For example : you pick up a card from a deck

E: P(Heart)= ¼

Not E: P(not Heart)= ¾

the events of one outcome happening

and that outcomes not happening

are complementary (opposite)

( not E is contrary event to E )

E Not E

For example : you pick up a card from a deck

E: P(Heart)= ¼

Not E: P(not Heart)= ¾

3.
Complementary Events

THE SUM of the of the probabilities of

complementary events is 1.

Not E

from which I get:

: Probability of the contrary event Not E is:

“1 minus the Probability of the event E”

Not E

THE SUM of the of the probabilities of

complementary events is 1.

Not E

from which I get:

: Probability of the contrary event Not E is:

“1 minus the Probability of the event E”

Not E

4.
Complementary Events

You pick up a card from deck of 52 cards.

Which is the probability of picking a figures?

12

P( figures) =

52

Which is the probability of not picking a

figures ?

12 52 − 12 40

P(NotFigures) = 1− = =

52 52 52

You pick up a card from deck of 52 cards.

Which is the probability of picking a figures?

12

P( figures) =

52

Which is the probability of not picking a

figures ?

12 52 − 12 40

P(NotFigures) = 1− = =

52 52 52

5.
Complementary Events

1) The probability that it will rain tomorrow is 0.4 .

What is the probability it does not rain ?

P(not Rain) = 1-0.4 = 0.6

2) Tossing 2 coins,which is the probability of:

a) never getting Tail ?

P(never T) = P(Head Head) = 1/4

b) getting at least once Tail?

(TT or HT or TH )

P(at Least Once T)=1-P(never T)=1-1/4= 3/4

1) The probability that it will rain tomorrow is 0.4 .

What is the probability it does not rain ?

P(not Rain) = 1-0.4 = 0.6

2) Tossing 2 coins,which is the probability of:

a) never getting Tail ?

P(never T) = P(Head Head) = 1/4

b) getting at least once Tail?

(TT or HT or TH )

P(at Least Once T)=1-P(never T)=1-1/4= 3/4

6.
ADDITION RULE

PROBABILITY OF

A OR B

PROBABILITY OF

A OR B

7.
1) Disjoint Events

There are two situations

2) NOT Disjoint Events

Disjoint Events ?

Two events are Disjoint ( Mutually Exclusive ) if they

can't happen at the same time

Turning left and turning right are Mutually Exclusive (you can't do

both at the same time)

Cards: Kings and Aces are disjoint

What is Not Disjoint ( not Mutually Exclusive ) ?

Turning left and scratching your head can happen at the same time

Cards: Kings and Hearts, because we can have a King of Hearts!

There are two situations

2) NOT Disjoint Events

Disjoint Events ?

Two events are Disjoint ( Mutually Exclusive ) if they

can't happen at the same time

Turning left and turning right are Mutually Exclusive (you can't do

both at the same time)

Cards: Kings and Aces are disjoint

What is Not Disjoint ( not Mutually Exclusive ) ?

Turning left and scratching your head can happen at the same time

Cards: Kings and Hearts, because we can have a King of Hearts!

8.
1) DISJOINT Events (Mutually Exclusive )

A single card is chosen at random from a

standard deck of 52 playing cards.

What is the probability of choosing an Ace

or a King?

P(ACE or KING ) = P(Ace) + P(King)

= 4/52 + 4/52 = 8/52

A single card is chosen at random from a

standard deck of 52 playing cards.

What is the probability of choosing an Ace

or a King?

P(ACE or KING ) = P(Ace) + P(King)

= 4/52 + 4/52 = 8/52

9.
1) DISJOINT events

Addition Rule B

A

for DISJOINT

Events:

When two events A and B are disjoint, the

probability that A or B will occur is:

the SUM of the Probability of each Event.

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

Addition Rule B

A

for DISJOINT

Events:

When two events A and B are disjoint, the

probability that A or B will occur is:

the SUM of the Probability of each Event.

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

10.
2) NOT DISJOINT ( NOT Mutually Exclusive)

example

A single card is chosen at random from a

standard deck of 52 playing cards.

What is the probability

of choosing

an Heart or a King?

P(H or K) = P(H) + P(K) - P(both)

=13/52 + 4/52 –1/52 = 16/52

example

A single card is chosen at random from a

standard deck of 52 playing cards.

What is the probability

of choosing

an Heart or a King?

P(H or K) = P(H) + P(K) - P(both)

=13/52 + 4/52 –1/52 = 16/52

11.
A and B = intersection

1) NOT DISJOINT

ADDITION RULE

for NOT disjoint A or B = union

Events

When two events A and B are NOT DISJOINT,

the probability that A or B will occur is :

the SUM of the probability of each event,

MINUS the probability of the overlap.

both

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

U union ∩ Intersection

1) NOT DISJOINT

ADDITION RULE

for NOT disjoint A or B = union

Events

When two events A and B are NOT DISJOINT,

the probability that A or B will occur is :

the SUM of the probability of each event,

MINUS the probability of the overlap.

both

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

U union ∩ Intersection

12.
SUMMARY : ADDITION RULE

DISJOINT EVENTS NOT DISJOINT EVENTS

Mutually Exclusive Not Mutually Exclusive

A and B together is impossible:

P(A and B) = 0 A and B together is possible !

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

DISJOINT EVENTS NOT DISJOINT EVENTS

Mutually Exclusive Not Mutually Exclusive

A and B together is impossible:

P(A and B) = 0 A and B together is possible !

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

13.
TRY IT YOURSELF

TEST

1: A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or a figure?

2: A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or Red Card?

3: You are going to roll two dice. Find:

P(sum that is even or sum that is a multiple of 3).

TEST

1: A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or a figure?

2: A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or Red Card?

3: You are going to roll two dice. Find:

P(sum that is even or sum that is a multiple of 3).

14.
1: A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or a figure?

Figures

P(Ace)=4/52 Aces

These events are mutually exclusive ( disjoint)

since they cannot occur at the same time.

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

U union

P(Ace OR Figure) = 4/52+12/52 = 16/52

standard deck of 52 playing cards. What is the

probability of choosing an Ace or a figure?

Figures

P(Ace)=4/52 Aces

These events are mutually exclusive ( disjoint)

since they cannot occur at the same time.

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

U union

P(Ace OR Figure) = 4/52+12/52 = 16/52

15.
2. A single card is chosen at random from a

standard deck of 52 playing cards. What is the

probability of choosing an Ace or Red Card?

Aces Red cards

P(Red card)=26/52

These events are NOT disjoint since they have

some overlap ( favorable outcomes in common )

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

U union ∩ Intersection

P(Ace OR Red Card) = 4/52+26/52-2/52 = 28/52

standard deck of 52 playing cards. What is the

probability of choosing an Ace or Red Card?

Aces Red cards

P(Red card)=26/52

These events are NOT disjoint since they have

some overlap ( favorable outcomes in common )

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

U union ∩ Intersection

P(Ace OR Red Card) = 4/52+26/52-2/52 = 28/52

16.
ANSWER 3

3. You are going to roll two dice. Find

P(sum that is even or sum that is a multiple of 3).

The addition rule says we need to find

P(even) + P(multiple of 3) - P(both)

The number of possible outcomes of rolling two dice = 36

P(even) means how many ways to roll:2, 4, 6, 8, 10, or 12.

P(even) = 18/36

P(multiple of 3) means how many ways to roll : 3, 6, 9 or 12.

P(multiple of 3) = 12/36

P(both) means what is the overlap. Notice that 6 and 12

occur in both places and have been counted twice. We need

to subtract those out. P(both) = 6/36

P(even or multiple of 3)= 18/36 + 12/36 - 6/36 = 24/36

3. You are going to roll two dice. Find

P(sum that is even or sum that is a multiple of 3).

The addition rule says we need to find

P(even) + P(multiple of 3) - P(both)

The number of possible outcomes of rolling two dice = 36

P(even) means how many ways to roll:2, 4, 6, 8, 10, or 12.

P(even) = 18/36

P(multiple of 3) means how many ways to roll : 3, 6, 9 or 12.

P(multiple of 3) = 12/36

P(both) means what is the overlap. Notice that 6 and 12

occur in both places and have been counted twice. We need

to subtract those out. P(both) = 6/36

P(even or multiple of 3)= 18/36 + 12/36 - 6/36 = 24/36