Basic Probability

1

• ????????????????????
• Basic Probability

2
Goals

• ???????????????????????
• ??????????????????????????????????????????????
• ??? contingency tables ?????????????????
• ??????????????????????????????????????????????????
  ??????
• ??????????????????????????????????
• ????????????????????????????????????????????????
• ????????????????????????????? (Bayes Theorem)
  ????????????????????????????????

3
Sample Spaces and Events

• ??????????????? (Random Experiments)

4
Sample Spaces and Events

• ??????????????? (Random Experiments)
• Definition
• ?????????????????? ? ?????????????????????????????
  ?????????????????????? ???? ? ????????????????????
  ???????????????????

5
Sample Spaces and Events

• Sample Spaces
• Definition

6
Sample Space
???????? ????????????? 1 ??? 1 ?????
?????????????????????????????? 6 ???
??????????? 1 ?? ?????? 1 ?????
????????????????????? 52 ????
7
Sample Spaces and Events

• ???????????????? Sample Space

?????????????????????????? ???????????????????????
?????????????????????? ??????????????????????????
????????????????????????????? ? ????????????????
??????????? ??? ??????????????????????????????
???????????????????????????????????????? ?????????
???????????
8
Example (continued)
9
Example (continued)
10
Example (continued)
11
Sample Spaces

• Tree Diagrams
• Sample spaces ????????????????? tree diagrams.
• ?????sample space ?????????????????????????? ?
  ??? ??????????????????????????????????????????????
  ???? 1 ???? n1 ?????????????????????????
  ????????????? n1 ????
• ??????????????????????????????????????????????????
  2 ???? n2 ?????????????????????????
  ????????????? n2 ????
• .

12
Sample Spaces

• Example 2 ????????????????????? 3
  ???????????????? ?????????????????????????????????
  ??? Late ??? On time ?????

13
Events

• Simple event
• ???????????? Sample Space ????????????????????????
  
• ???? ??? red card ?????? 1 ?????
• Complement ???????????? A (??????? A)
• ??????????????????????????????????? A
• ???? ??????????????????????? diamonds
• ????????????? (Joint event)
• ??????????? ? ?????????????????????????? 2
  ?????????? ? ???
• ???? ??? ace ????????? ??? ????????????????

14
Visualizing Events

• Contingency Tables
• Tree Diagrams

Ace Not Ace Total
Black 2 24 26
Red 2 24
26
Total 4 48
52
Sample Space
2 24 2 24
Ace
Sample Space
Black Card
Not an Ace
Full Deck of 52 Cards
Ace
Red Card
Not an Ace
15
Mutually Exclusive Events

• Mutually exclusive events
• ????????????????????????????
• example
• A ??? Queen ????? B ??? Queen ????
• Events A ??? B ????????????? mutually exclusive

16
Collectively Exhaustive Events

• ????????????
• ??????????? ? ??????????????
• ???????????????????????????????? Sample Space
• example ????????????????????
• A Ace B ????
• C ??????????? D ????????
• Events A, B, C ??? D ????????????????
  collectively exhaustive (?????? mutually
  exclusive)
• Events B, C ??? D ???????????????? collectively
  exhaustive

17
Sample Spaces and Events

• Basic Set Operations

18
Sample Spaces and Events

• Venn Diagrams

19
Sample Spaces and Events

• Definition

20
Probability

• ??????????????????????????????????????????????????
  ???????????????? ?
• ???????????? 0 ??? 1
• ????????????????? mutually exclusive ???
  collectively exhaustive ?????????????? 1

Certain
1
0 P(A) 1 For any event A
.5
Impossible
0
????? A, B, and C ??? mutually exclusive
???collectively exhaustive
21
Assessing Probability

• Approaches to assessing the probability of un
  uncertain event
• 1. a priori classical probability
• 2. empirical classical probability

22
2-2 Interpretations of Probability
Definition
The notations may varies depend on the types of
books
23
Interpretations of Probability
Example 3
24
Interpretations of Probability
?????????????????????????
25
Addition Rules
Addition Rule????????
Mutually Exclusive Events
26
Addition Rules
Three or More Events
27
Addition Rules
Venn diagram of four mutually exclusive events
28
Addition Rules
29
Computing Probabilities

• The probability of a joint event, A and B
• Computing a marginal (or simple) probability
• Where B1, B2, , Bk are k mutually exclusive and
  collectively exhaustive events

30
Joint Probability Example
P(Red and Ace)
Color
Type
Total
Black
Red
2
2
4
Ace
24
24
48
Non-Ace
26
26
52
Total
31
Marginal Probability Example
P(Ace)
Color
Type
Total
Black
Red
2
2
4
Ace
24
24
48
Non-Ace
26
26
52
Total
32
Joint Probabilities Using Contingency Table
Event
Total
B1
B2
Event

P(A1 and B2)
P(A1)
P(A1 and B1)
A1
P(A2 and B1)
A2
P(A2 and B2)
P(A2)
Total
1
P(B1)
P(B2)
Marginal (Simple) Probabilities
Joint Probabilities
33
General Addition Rule Example
P(Red or Ace) P(Red) P(Ace) - P(Red and Ace)
26/52 4/52 - 2/52
28/52
Dont count the two red aces twice!
Color
Type
Total
Black
Red
2
2
4
Ace
24
24
48
Non-Ace
26
26
52
Total
34
Conditional Probability

• ?????????????????????? ????????????????????? D
  ??????????????????????????? ??? F
  ?????????????????????????????????????
• ??????????????????????????????????????????????????
  ???????????????? (E)
• ????????????????????? E ???? P(DF)
  ???????????????????????????????????? D given F
• ??????????????????????????????????????????????????
  ??????????????????????

35
Conditional Probability
Conditional probabilities for parts with surface
flaws
36
Conditional Probability
Definition
37
Computing Conditional Probabilities

• A conditional probability is the probability of
  one event, given that another event has occurred

The conditional probability of A given that B has
occurred
The conditional probability of B given that A has
occurred
Where P(A and B) joint probability of A and B
P(A) marginal probability of A P(B)
marginal probability of B
38
Conditional Probability Example

• Of the cars on a used car lot, 70 have air
  conditioning (AC) and 40 have a CD player (CD).
  20 of the cars have both.

• What is the probability that a car has a CD
  player, given that it has AC ?
• i.e., we want to find P(CD AC)

39
Conditional Probability Example
(continued)

• Of the cars on a used car lot, 70 have air
  conditioning (AC) and 40 have a CD player (CD).
  
• 20 of the cars have both.

No CD
CD
Total
.2
.5
.7
AC
.2
.1
No AC
.3
.4
.6
1.0
Total
40
Conditional Probability Example
(continued)

• Given AC, we only consider the top row (70 of
  the cars). Of these, 20 have a CD player. 20
  of 70 is about 28.57.

No CD
CD
Total
.2
.5
.7
AC
.2
.1
No AC
.3
.4
.6
Total
1.0
41
Using Decision Trees
P(AC and CD) .2
Given AC or no AC
Has CD
P(AC) .7
Does not have CD
P(AC and CD) .5
Has AC
All Cars
Does not have AC
P(AC and CD) .2
Has CD
P(AC) .3
Does not have CD
P(AC and CD) .1
42
Using Decision Trees
(continued)
P(CD and AC) .2
Given CD or no CD
Has AC
P(CD) .4
Does not have AC
P(CD and AC) .2
Has CD
All Cars
Does not have CD
P(CD and AC) .5
Has AC
P(CD) .6
Does not have AC
P(CD and AC) .1
43
Statistical Independence

• Two events are independent if and only if
• Events A and B are independent when the
  probability of one event is not affected by the
  other event

44
Multiplication Rules

• Multiplication rule for two events A and B

Note If A and B are independent, then
and the multiplication rule simplifies to
45
Total Probability Rules
Partitioning an event into two mutually exclusive
subsets.
Partitioning an event into several mutually
exclusive subsets.
46
Total (marginal) Probability Rules
47
Total Probability Rules
Example 4
48
Total Probability Rules
multiple events
49
Independence
Definition
50
Independence
Definition
51
Example 5
52
Bayes Theorem

• where
• Bi ith event of k mutually exclusive and
  collectively
• exhaustive events
• A new event that might impact P(Bi)

53
??????????? (Permutations)

• ?????????????????? ???????????????????????????????
  ?????????????????????????????????????????? ?
  ????????????????????????? nPr (??????? n-P-r)
• ??????????? ??? n ???? ?????????????????????? n1,
  n2,, nk ???? ?????????????????????????
  ??????????????????

nPr
????? r n
????? ni lt n ??? n1 n2 nk n
54
?????????????? (Combinations)

• ?????????????? ??????? ???????????????????????????
  ??????????????????????? ? ????????????????????????
  ??? ??????????????????????????????????????????????
  ???????????????????????? ?????????????????????????
  nCr (??????? n-C-r) ??????????????????

nCr
????? r n
nCr