SECTION 11-2

• Events Involving “Not” and “Or”

Slide 11-2-1

EVENTS INVOLVING “NOT” AND “OR”

• Properties of Probability• Events Involving “Not”• Events Involving “Or”

Slide 11-2-2

PROPERTIES OF PROBABILITY

Slide 11-2-3

Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold.

1. 0 ( ) 1

2. ( ) 0

3. ( ) 1

P E

P

P S

(The probability of an event is between 0 and 1, inclusive.)

(The probability of an impossible event is 0.)

(The probability of a certain event is 1.)

EXAMPLE: ROLLING A DIE

Slide 11-2-4

When a single fair die is rolled, find the probability of each event.

a) the number 3 is rolledb) a number other than 3 is rolledc) the number 7 is rolledd) a number less than 7 is rolled

EXAMPLE: ROLLING A DIE

Slide 11-2-5

SolutionThe outcome for the die has six possibilities: {1, 2, 3, 4, 5, 6}.

1a) (3)

65

b) (not 3)6

c) (7) 0

d) (less than 7) 1

P

P

P

P

EVENTS INVOLVING “NOT”

Slide 11-2-6

The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction.

CORRESPONDENCES

Slide 11-2-7

Set Theory Logic Arithmetic

Operation or Connective (Symbol)

Complement Not Subtraction

Operation or Connective (Symbol)

Union Or Addition

Operation or Connective (Symbol)

Intersection And Multiplication

( )

( ) ( )

( ) ( )

( )

( ) ( ) ( )

PROBABILITY OF A COMPLEMENT

Slide 11-2-8

The probability that an event E will not occur is equal to one minus the probability that it will occur.

(not ) ( ) ( )

1 ( )

P E P S P E

P E

S

( ) 1

( ) 1 ( ).

P E P E

P E P E

So we have

EE

and

EXAMPLE: COMPLEMENT

Slide 11-2-9

When a single card is drawn from a standard 52-card deck, what is the probability that it will not be an ace?

Solution(not an ace) 1 (ace)

4 1

5248 12

.52 13

P P

EVENTS INVOLVING “OR”

Slide 11-2-10

Probability of one event or another should involve the union and addition.

MUTUALLY EXCLUSIVE EVENTS

Slide 11-2-11

Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.)

ADDITION RULE OF PROBABILITY (FOR A OR B)

Slide 11-2-12

If A and B are any two events, then

( or ) ( ) ( ) ( and ).P A B P A P B P A B If A and B are mutually exclusive, then

( or ) ( ) ( ).P A B P A P B

EXAMPLE: PROBABILITY INVOLVING “OR”

Slide 11-2-13

When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or a diamond?

Solution(king or diamond) (K) (D) (K and D)

4 13 1

52 52 5216 4

.52 13

P P P P

EXAMPLE: PROBABILITY INVOLVING “OR”

Slide 11-2-14

If a single die is rolled, what is the probability of a 2 or odd?

Solution

(2 or odd) (2) (odd)

1 3 4 2 .

6 6 6 3

P P P

These are mutually exclusive events.