This ppt is prepared for engineering class. Random variable and stochastic prosses and probability.

1. Review of Las Lecture
• We introduced the subject
• Probability was discussed
• Single event was introduced
• Any Problem??

2. Lecture 2
Today we will study about….
1.3 Combinations of Events

3. 1.3 Combinations of Events
1.3.1 Intersections of Events(1/5)
• Intersections of Events
The event is the intersection of the events A and B and
consists of the outcomes that are contained within both events A
and B. The probability of this event, , is the probability that
both events A and B occur simultaneously.
A B∩
( )P A B∩

4. 1.3.1 Intersections of Events(2/5)
• A sample space consists of 9 outcomesS
( ) 0.01 0.07 0.19 0.27
( ) 0.07 0.19 0.04 0.14 0.12 0.56
( ) 0.07 0.19 0.26
P A
P B
P A B
= + + =
= + + + + =
∩ = + =

5. 1.3.1 Intersections of Events(3/5)
( ) 0.04 0.14 0.12 0.30
( ) 0.01
( ) ( ) 0.26 0.01 0.27 ( )
( ) ( ) 0.26 0.30 0.56 ( )
P A B
P A B
P A B P A B P A
P A B P A B P B
′∩ = + + =
′∩ =
′∩ + ∩ = + = =
′∩ + ∩ = + = =

6. 1.3.1 Intersections of Events(4/5)
• Mutually Exclusive Events
Two events A and B are said to be mutually exclusive if
so that they have no outcomes in common.
( ) ( ) ( )
( ) ( ) ( )
P A B P A B P A
P A B P A B P B
′∩ + ∩ =
′∩ + ∩ =
g
A B∩ = ∅

7. 1.3.1 Intersections of Events(5/5)
A B A∩ =
( ) ( )
A B B A
A A A
A S A
A
A A
A B C A B C
∩ = ∩
∩ =
∩ =
∩∅ = ∅
′∩ = ∅
∩ ∩ = ∩ ∩
g

8. 1.3.2 Unions of Events(1/4)
• Unions of Events
The event is the union of events A and B and consists of the
outcomes that are contained within at least one of the events A and
B. The probability of this event, , is the probability that at
least one of the events A and B occurs.
A B∪
( )P A B∪

9. 1.3.2 Unions of Events(2/4)
• Notice that the outcomes in the event can be classified into
three kinds.
1. in event but not in event
2. in event but not in event or
3. in both events and
A B∪
A
A
A
B
B
B
( ) ( ) ( ) ( )P A B P A B P A B P A B′ ′∪ = ∩ + ∩ + ∩

10. 1.3.2 Unions of Events(3/4)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
If the events and are mutually exclusive so that
( ) 0, then ( ) ( ) ( )
P A B P A P A B
P A B P B P A B
P A B P A P B P A B
A B
P A B P A B P A P B
′∩ = − ∩
′∩ = − ∩
∪ = + − ∩
∩ = ∪ = +
g
g
g

11. 1.3.2 Unions of Events(4/4)
• Simple results concerning the unions of events
( )
( )
( ) ( )
A B A B
A B A B
A B B A
A A A
A S S
A A
A A S
A B C A B C
′ ′ ′∪ = ∩
′ ′ ′∩ = ∪
∪ = ∪
∪ =
∪ =
∪∅ =
′∪ =
∪ ∪ = ∪ ∪

12. 1.3.3 Examples of Intersections and Unions(1/11)
• Example 5 Television Set Quality
A company that manufactures television sets performs a final quality
check on each appliance before packing and shipping it.
The quality check has an evaluation of the quality of the picture and
the appearance. Each of the two evaluations is graded as Perfect
(P), Good (G), Satisfactory (S), or Fail (F).

13. 1.3.3 Examples of Intersections and Unions(2/11)
An appliance that fails on either of the two evaluations and that score
an evaluation of Satisfactory on both accounts will not be shipped.
A = { an appliance cannot be shipped }
= { (F,P), (F,G), (F,S), (F,F), (P,F), (G,F), (S,F), (S,S) }
P(A) = 0.074
About 7.4% of the television
sets will fail the quality check.

14. 1.3.3 Examples of Intersections and Unions(3/11)
B = { picture satisfactory
or fail }
P(B) =
0.178

15. 1.3.3 Examples of Intersections and Unions(4/11)
= { Not shipped and the picture satisfactory or fail }A B∩
( ) 0.055P A B∩ =

16. 1.3.3 Examples of Intersections and Unions(5/11)
= { the appliance was either not shipped or the picture
was evaluated as being either Satisfactory or Fail }
A B∪
( ) 0.197P A B∪ =

17. 1.3.3 Examples of Intersections and Unions(6/11)
= { Television sets that have a picture evaluation of either
Perfect or Good but that cannot be shipped }
( ) 0.019
Notice
( ) ( ) 0.055 0.019
0.074
( )
P A B
P A B P A B
P A
′∩ =
′∩ + ∩ = +
=
=
( ')P A B∩

18. 1.3.3 Examples of Intersections and Unions(7/11)
• GAMES OF CHANCE
- A = { an even score is obtained from a roll of a die }
= { 2, 4, 6 }
B = { 4, 5, 6 }
then
- A = { the sum of the scores is equal to 6 }
B = { at least one of the two dice records a 6 }
P(A) = 5/36 and P(B) = 11/36
=> A and B are mutually exclusive
{4,6} and {2,4,5,6}
2 1 4 2
( ) and ( )
6 3 6 3
A B A B
P A B P A B
∩ = ∪ =
∩ = = ∪ = =
and ( ) 0A B P A B∩ = ∅ ∩ =
16 4
( ) ( ) ( )
36 9
P A B P A P B∪ = = = +

19. 1.3.3 Examples of Intersections and Unions(8/11)
- One die is red and the other is blue (red, blue).
A = { an even score is obtained on the red die }

20. 1.3.3 Examples of Intersections and Unions(9/11)
B = { an even score is obtained on the blue die }

21. 1.3.3 Examples of Intersections and Unions(10/11)
= { both dice have even scores }A B∩
9 1
( )
36 4
P A B∩ = =

22. 1.3.3 Examples of Intersections and Unions(11/11)
= { at least one die has an even score }A B∪
27 3
( )
36 4
P A B∪ = =

23. 1.3.4 Combinations of Three or More Events(1/4)
• Union of Three Events
The probability of the union of three events A, B, and C is the sum
of the probability values of the simple outcomes that are contained
within at least one of the three events. It can also be calculated from
the expression
[ ]
[ ]
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
P A B C P A P B P C
P A B P A C P B C
P A B C
∪ ∪ = + +
− ∩ + ∩ + ∩
+ ∩ ∩

24. 1.3.4 Combinations of Three or More Events(2/4)
( ) ( ) ( ) ( )P A B C P A P B P C∪ ∪ = + +

25. 1.3.4 Combinations of Three or More Events(3/4)
• Union of Mutually Exclusive Events
For a sequence of mutually exclusive events, the
probability of the union of the events is given by
• Sample Space Partitions
A partition of a sample space is a sequence
of mutually exclusive events for which
Each outcome in the sample space is then contained within one
and only one of the events
1 2, ,..., nA A A
1 1( ) ( ) ( )n nP A A P A P A∪ ∪ = + +L L
1 2, ,..., nA A A
1 nA A S∪ ∪ =L
iA

26. 1.3.4 Combinations of Three or More Events(4/4)
• Example 5 Television Set Quality
C = { an appliance is of “mediocre quality” }
= { score either Satisfactory or Good }
= { (S,S), (S,G), (G,S), (G,G) }
D = { an appliance is of “ high quality” }
=
= { (G,P), (P,P), (P,G), (P,S) }
P(D) = 0.523
( )A B C A B C′ ′ ′ ′∪ ∪ = ∩ ∩

27. Home Take Assignment
• Solve the End exercise problems relevant
to the today class and last lecture.