Download mutually exclusive events

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
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  P( E )  1
(The probability of an event is
between 0 and 1, inclusive.)
2. P()  0
(The probability of an impossible
event is 0.)
3. P( S )  1
(The probability of a certain event
is 1.)
Slide 11-2-3
EXAMPLE: ROLLING A DIE
When a single fair die is rolled, find the
probability of each event.
a) the number 3 is rolled
b) a number other than 3 is rolled
c) the number 7 is rolled
d) a number less than 7 is rolled
Slide 11-2-4
EXAMPLE: ROLLING A DIE
Solution
The outcome for the die has six possibilities:
{1, 2, 3, 4, 5, 6}.
1
a) P (3) 
6
5
b) P (not 3) 
6
c) P (7)  0
d) P (less than 7)  1
Slide 11-2-5
EVENTS INVOLVING “NOT”
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.
Slide 11-2-6
CORRESPONDENCES
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
( )
()
()
Slide 11-2-7
PROBABILITY OF A COMPLEMENT
The probability that an event E will not occur is
equal to one minus the probability that it will occur.
P(not E )  P( S )  P( E )
E
S
E
 1  P( E )
So we have
P( E )  P  E    1
and P( E )  1  P( E ).
Slide 11-2-8
EXAMPLE: COMPLEMENT
When a single card is drawn from a standard 52-card
deck, what is the probability that it will not be an ace?
Solution
P (not an ace)  1  P(ace)
4
1
52
48 12

 .
52 13
Slide 11-2-9
EVENTS INVOLVING “OR”
Probability of one event or another should
involve the union and addition.
Slide 11-2-10
MUTUALLY EXCLUSIVE EVENTS
Two events A and B are mutually exclusive events
if they have no outcomes in common. (Mutually
exclusive events cannot occur simultaneously.)
Slide 11-2-11
ADDITION RULE OF PROBABILITY (FOR A
OR B)
If A and B are any two events, then
P( A or B)  P( A)  P( B)  P( A and B).
If A and B are mutually exclusive, then
P( A or B)  P( A)  P( B).
Slide 11-2-12
EXAMPLE: PROBABILITY INVOLVING
“OR”
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
P (king or diamond)  P(K)  P(D)  P(K and D)
4
13
1



52
52
52
16 4

 .
52 13
Slide 11-2-13
EXAMPLE: PROBABILITY INVOLVING
“OR”
If a single die is rolled, what is the probability of a
2 or odd?
Solution
These are mutually exclusive events.
P(2 or odd)  P(2)  P(odd)
1

6
3

6
4 2
  .
6 3
Slide 11-2-14