Experiment and random phenomenon

1
Experiment and random phenomenon
??

• An experiment is any activity from which an
  outcome, measurement, or result is obtained.
  ????????????????????
• When the outcomes cannot be predicted with
  certainty, the experiment is a random experiment.

2
Randomness???

• ?????????????????????????????,?????????????,??????
  ???????
• ??????????????????????????????,???????????,???????
  ????

3
Example of Experiments
??

• ??
• Measuring the lifetime (time to failure) of a
  given product
• Inspecting an item to determine whether it is
  defective
• ???
• ????
• ????
• ????????

4

• ?????????????????????????

5
Probability??

• ??????????????????????
• ????????????????????????,???????????????,????,???
  ????????

6
Probability Models????

• ??????????
• (1) ???????????????(outcomes)
• (2) ????????????

7
Basic Outcomes and Sample Space
??

• The set of all possible basic outcomes for a
  given experiment (random phenomenon) is called
  the sample space.?????????????????,???S?O???
• Each possible outcome of a random experiment is
  called a basic outcome (or a sample point, an
  element in the sample space). ????????????????????
  ???????(??????????),???oi???

8
Basic Outcomes and Sample Space
??

• ????????????????,????????????????????,?????????
• o1 ?? o2 ?? o3 ??
• o4 ?? o5 ?? o6 ??
• ????????
• S o1 , o2 , o3 , o4 ,o5 ,o6
• ??????????????,?????????? o4 ??,?????o4??????

9
Venn Diagrams
??

• S o1 , o2 , o3 , o4 ,o5 ,o6

?
?
?
o1 o2 o3 o4 o5 o6
?
?
?
10
Event??
??

• An event is an outcome or a set of outcomes of a
  random phenomenon. That is, an event is a subset
  of the sample space.
• ?????????????????????(????)????,????????????(subse
  t),????????????

11
Event??
??
??

• ???,??????????????
• A o4 ,o5 ,o6
• ?????????????,?????A??????
• ?B???????????????,?
• B o1 ,o5

12
Venn Diagrams
??

• S o1 , o2 , o3 , o4 ,o5 ,o6

B??
?
?
?
o1 o2 o3 o4 o5 o6
?
?
?
A??
13
Event??
??

• ????????????????
• S 1, 2, 3, 4, 5, 6
• ?????????
• A 2,4,6
• ??????2???
• B 3,4,5,6

14
Event??
??

• ?????H???T??,?????????????
• S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
• ?A????????????,?
• A HHH, HHT, HTH, HTT

15
Assigning Probabilities to Events?????
??

• There are two types of random experiments, those
  that can be repeated over and over again under
  essentially identical conditions and those that
  are unique and cannot be repeated.

16
Assigning Probabilities to Events?????
??

• A numerical measure that indicates the likelihood
  of a specific outcome in a repeatable random
  experiment is called an objective probability,
  whereas the probability associated with a
  specific outcome of a unique and nonrepeatable
  random experiment is called a subjective
  probability.

17
Assigning Probabilities to Events?????
??

• There are three different approaches to assigning
  probabilities to basic outcomes
• 1. The relative frequency approach
• 2. The equally likely approach
• 3. The subjective approach

18
The Relative Frequency Approach ????(??)????
??

• Let fA be the number of occurrences, or frequency
  of occurrence, of event A in n repeated identical
  trials. The probability that A occurs is the
  limit of the ratio fA/n as the number of trials n
  becomes infinitely large.

19
The Relative Frequency Approach ????(??)????
??

• ?????????????????,???A?????????????????????????,A
  ???????????????
• ??????????????,??????????????????????????????????,
  ??????????????

20
?????????
??

• (1) Because we can never replicate an experiment
  an infinite number of times, it is impossible to
  determine the limit of the ratio fA/n as n
  approaches infinity.
• (2) We can never be sure that we have repeated an
  experiment under identical conditions.

21
?????????
??

• When we use the relative frequency approach, we
  use the observed ratio fA/n to approximate the
  theoretical probability that event A occurs. That
  is, we assume that P(A) ? fA/n when n is
  sufficiently large.

22
The Equally Likely Approach??(??)????
??

• Suppose that an experiment must result in one of
  n equally likely outcomes. Then each possible
  basic outcome is considered to have probability
  1/n of occurring on any replication of the
  experiment.
• ??????n????(????),?????????????????????????,??????
  ??????1/n????A???????nA,?A??????
• P(A) nA/n,??????????????????

23
The Equally Likely Approach??(??)????
??

• ??????,??????6????
• E (1,5) (2,4) (3,3) (4,2) (5,1)
• P(E) 5/36
• ?????????????????
• ?????????????????????????

24
Objective Probability????
??

• A probability obtained by using a relative
  frequency approach or an equally likely approach
  is called an objective probability.

25
The Subjective Approach????
??

• ?????????????,???????????????????,???????,????????
  ?,?????????????
• ?????,??????????,???????????????,??????????????
• A subjective probability is a number in the
  interval 0, 1 that reflects a person's degree
  of belief that an event will occur.

26
Odds???
??

• ???????????????????(odds)?????????????????
• ???????????????????31???,??????75?????
• If the odds in favor of event A occurring are a
  to b, then

27
Which approach is best?
??

• The nature of the problem determines which
  approach is best.
• Problems with an underlying symmetry, such as
  coin, dice, and card problems, are especially
  suited to the equally likely approach.
• Problems for which we have large samples of data
  based on many replications of an experiment are
  especially suited to the relative frequency
  approach.
• Problems that occur only once, such as a sporting
  event, are especially suited to the subjective
  approach.

28
Which approach is best?
??

• ?????,????????
• (1) ?????
• ???????,???????1/2
• (2) ?????
• ?????????52,???????52
• (3) ????
• ?????????,?????????????,?????????60

29
Set Theory
??

• Subset???
• An event A is contained in another event B if
  every outcome that belongs to the subset defining
  the event A also belongs to the subset defining
  the event B.
• A 2,4,6 B2,3,4,5,6
• A ? B, A is a subset of B
• If A ? B and B ? A, then A B
• If A ? B and B ? C, then A ? C
• Empty Set or Null Set???Ø
• For any event A, Ø ? A ? S,

30
Operation of Set Theory Unions??
??
S

• Unions??
• Let A and B be two events in the sample space S.
  Their union, denoted A U B. is the event composed
  of all basic outcomes in S that belong to at
  least one of the two events A or B. Hence, the
  union A U B occurs if either A or B (or both)
  occurs.

A
B
31
Operation of Set Theory Unions??
??
S

• Unions?? The union of n events A1,A2,,An is
  defined to be the event that contains all
  outcomes which belong to at least one of these n
  events.

A
B
32
Operation of Set Theory Intersection??
??
S

• Intersection??
• Let A and B be two events in the sample space S.
  The intersection of A and B, denoted A ? B. is
  the event composed of all basic outcomes in S
  that belong to both A and B. Hence, the
  intersection A ? B occurs if both A and B
  occur.

A
B
33
Operation of Set Theory Intersection??
??
S

• Intersection??
• The intersection of n events, A1, An is defined
  to be the event that contains the outcomes which
  are common to all these n events.

A
B
34
Complement of an Event
??

• Let A denote some event in the sample space S.
  The complement of A (A????), denoted by Ac,
  represents the event composed of all basic
  outcomes in S that do not belong to A.

S
A
Ac
35
Complement has the following properties
??

• (Ac)c A
• A ? Ac S
• Øc S
• Sc Ø
• A ? Ac Ø

S
A
Ac
36
Complement has the following properties
??

• (A ? B)c Ac nBc
• (A n B)c Ac ? Bc

S
Bc
B
A
Ac

• P(A) P(A n B) P(A n Bc)
• P(Ac n Bc) 1 - P(A ? B)
• P(Ac ? Bc) 1 - P(A n B)

37
Mutually Exclusive Events (Disjoint Events)

• Let A and B be two events in a sample space S. If
  A and B have no basic outcomes in common, then
  they are said to be mutually exclusive. If A and
  B are mutually exclusive events, we write (A ? B)
  Ø, where Ø denotes the empty set. P(A ? B) 0.
  

38
Some basic rules of probability
??

• Probability of a basic outcome
• For each basic outcome oi, 0 ?P(oi) ? 1.
• Probability of an event
• Let event A o1 , o2 , o3 , o4 ,o5 ,ok ,
  where o1 , o2 , o3 , o4 ,o5 ,ok are k different
  basic outcomes. The probability of any event A is
  the sum of the probabilities of the basic
  outcomes in A. That is,
• P(A) P(o1) P(o2) P(o3) P(o4) P(o5)
  P(ok) ?AP(oi)
• where ?AP(oi) means to obtain the sum over all
  basic outcomes in event A.

39
Some basic rules of probability
??

• Rule1. ??????????0?1??
• For each basic outcome oi, 0 ?P(oi) ? 1.
• The probability of P(A) satisfies 0 ?P(A) ? 1.
• Rule 2. ????????????????????
• Let event S o1 , o2 , o3 , o4 ,o5 ,on
  represent the sample space of an experiment. The
  probability of S is P(S) ?sP(oi) 1
• Rule 3. ????????????1???????
• P(Ac) 1- P(A)
• Rule 4. ????????????,?????????????????
• If A and B are disjoint P(A or B) P(A) P(B)

40
Definition of Probability

• Axiom 1??1 For any event A, P(A) ?
  0??A????????
• Axiom 2 P(S) 1.
• Axiom 3 For any infinite sequence of disjoint
  events (????) A1, A2,

41
Definition of Probability

• A probability distribution , or simply a
  probability, on a sample space S is a
  specification of numbers P(A) which satisfy
  Axioms 1,2, and 3.
• ???????????S,?S??????A????P(A),?P(.)????????,??P(.
  )??????,??P(A) ???A????

42
Theorem 1??????

• ?????????
• Pr(Ø)0

43
Theorem 2

• For any finite sequence of n disjoint events
  A1,A2,,An

???3????
44
Theorem 2

• Proof. ????????????A1, A2, A3,, ??A1 An ?????,
  Ai ?, i gt n

45
Theorem 3??????Probability of the complement of
an event
??

• Let Ac denote the complement of A. Then P(Ac) 1
  P(A).
• Proof
• Since A and Ac are disjoint events and A ? Ac
  S,
• it follows from Theorem 2 that P(S) P(A)
  P(Ac).
• Since P(S) 1 by Axiom 2,
• then P(Ac) 1 P(A).

46
Theorem 4?????
??

• For any event A, 0? P(A) ? 1.
• Proof.
• ???1?? P(A) ? 0.
• ?? P(A) gt 1
• ? P(Ac) lt 0 ?????1
• ?? P(A) ? 1

47
Theorem 5
??

• If A ? B, then P(A) ? P(B)
• Proof.
• B A ? BAc
• P(B) P(A) ? P(BAc )
• P(BAc ) ? 0
• P(A) ? P(B)

B
BAc
A
48
Theorem 6

• P(A ? B) P(A) P(B) P(AB)
• Proof
• P(A ? B) P(ABc) P(AB) P(AcB)
• P(A) P(ABc) P(AB)
• P(B) P(AcB) P(AB)

A
B
ABC
AB
ACB
49
Theorem 6

• P(A1 ? A2 ?A3)
• P(A1) P(A2) P(A3)
• P(A1 n A2 ) P(A2 n A3 )
• P(A1 n A3 ) P(A1 n A2 n A3 )

50
??

• Suppose that 15 of the freshmen fail chemistry,
• 12 fail math,
• and 5 fail both.
• Suppose a first-year student is picked at random.
  Find the probability that the student failed at
  least one of the courses.
• P(A ? B) P(A) P(B) P(AB)
• .15 .12 - .05 .22

51
????

• ?A, B???????????,?A?B??????????A?B????? (joint
  probability)?
• ?????A ?B?????????,?P(A nB)????

52
??

• ???????,
• A ?????
• B ???1
• P(A nB) 1/6

53
??

• ?????????,
• A ????7
• B ????????6
• P(AnB) 1/36

54
Joint Probability Tables?????
row sum
column sum
55
Joint Probability Tables?????
????????? 4700/12500 .376
A joint probability shows the probability that an
observation will possess two (or more)
characteristics simultaneously. Every joint
probability must be a number in the closed
interval 0,1 and the sum of all joint
probabilities must be 1.
56
Marginal Probability
????????68
P(??) P(????) P(????) .304 .128 .432
57
Marginal Probability
?????????????--?????????????????
58
Conditional Probability????
??

• P(A?B) The probability that some event A occurs
  given that some other event B has already
  occurred.
• If the probability of one event varies depending
  on whether a second event has occurred, the two
  events are said to be dependent.

59
Conditional Probability????

• A ?????
• B?????
• P(B) ???????? ? P(B ? A) ????????????

60
??

• ???????,
• A ?????
• B ???1
• P(B A) ?
• ????A??(?????),????????????????????????(reduced
  sample space)
• P(B A)(?????????B????????)/ (????????????)

61
Conditional Probability
??

• ????B???,??A???????
• (A?B???) ??(??B?????)

S
A
B
62
Conditional Probability

• ?????????????????
• P(?????) P(????????,??????)
• P(?????) P(????????,??????)

63
Conditional Probability
64
Conditional Probability
65
Conditional Probability
lt
??????????????????
66
Multiplicative law of probability
??
S
A
B
67
Multiplicative law of probability
??

• ?????????????60??????5?????????????????,?????????
  ??????
• A ?????????
• B ?????????
• AnB??????

68
Multiplicative law of probability
??

• ?????,????????????,???????,???????????
• A ?????????
• C ??????????
• P(CA) 55/59

69
Independence??
??

• ????A????????B??????,?????B???,?A?B?????
• Event A and B are independent if and only if

70
Independence??
??

• ?A?B???,?

• P(AB) P(A)

• P(BA) P(B)

71
Independence??

• ?A, B??????????,?P(A) ?0, P(B) ?0

• P(AB) P(A)

? P(BA) P(B)

• ?A, B?????????

• P(AB) ? P(A)

? P(BA) ? P(B)

• ?A, B?????????(dependent events)

72
??

• ???????,
• A ?????
• B ???1
• P(B A) 1/3 ? P(AB) 1

73
Independence??
??
????
??
????
74
?????

• ?A?B????,?A?B?????

• P(AB) P(A)

?P(A) ?0 ?P(AnB) ?0 AnB ?????,??A, B???
75
?????

• ?A?B?????,?A?B?????

???? P(AnB) ?0
P(AB) ? 0
P(AB) ?0?????? P(AB) P(A)
76
?????

• ?A?B????,?A?B?????

??? P(AnB) 0
P(AB) 0 ?P(A)
P(AB) ?P(A) ?? A?B???
???????B????,A??????,????????
77
?????

• ?A?B?????,?A?B?????

78
Independence??
??

• Theorem If two events A and B are independent,
  then the events A and Bc are also independent.
• Proof.

S
A
B
? A and B are independent
79
Independence??
??

• Approximately 30 of the sales representatives
  hired by a firm quit in less than 1 year. Suppose
  that two sales representatives are hired and
  assume that the first sales representative's
  behavior is independent of the second sales
  representative's behavior.
• (a) What is the probability that both quit within
  a year?
• (b) Find the probability that exactly one
  representative quits.

80
Independence??
??

• (a) What is the probability that both quit within
  a year?
• (b) Find the probability that exactly one
  representative quits.

81
Tree Diagrams
??
B
.3
A
.7
Bc
.3
B
.3
.7
Ac
.7
Bc
82
Independence??
??

• ??20?????????????????????????70????????????70????
  ?.6,????70?????.7,??????????????,??????????,??????
  ???????????

83
Sampling with and without replacement
??

• Selecting a random sample can be viewed as a
  process in which we sequentially obtain one
  observation after another.
• When we sample with replacement, successive
  outcomes are independent

When we sample without replacement, successive
outcomes are not independent
84
Sampling without replacement
??

• ?????????????????,???????????,????????,?????????,?
  ??????????
• ? A???????? B???????

85
Sampling with replacement
??

• ??????7????????,IRS??????,???????????????,????????
  ?????????

86
??????
??

• The sample space of an experiment is partitioned
  into k mutually exclusive and exhaustive events
  A1, A2, Ak
• ?A1, A2, Ak?????S?????,???????
• 1. A1?A2 ?A3 ?Ak S
• 2. AinAj ?
• ??A1, A2, Ak?????S????(partition)

87
??????
??

• ?A1, A2, Ak?????S????(partition),?B?S????????,
  ?A1B, A2B, AkB???B?????

S
A1
A2
B
A5
A3
A4
88
??????
??
S
A1
A2
B
A5
A3
A4

• ?A1, A2, Ak?????S????(partition),?P(Aj)gt0,????
  S????B

89
??????
??

• ??????????????,?????????.6 ,?????????.4??????,????
  ???????.8,????,???????????.3,????????????(??)
• B ???????
• P(?) .6 P(?).4
• P(B?).8 P(B?).3
• P(B) P(?) P(B?) P(?) P(B?)
• .6 .8 .4 .3 .6

90
Bayes Theorem????
??

• ?A1, A2, Ak?????S????(partition),?P(Aj)gt0,?B?S
  ??????,?P(B)gt0, ?for i1,k

S
A1
A2
B
A5
A3
A4
91
Bayes Theorem????
??
Posterior probability ????
Prior probability ????
92
Bayes Theorem????
??

• ????????????,????????????????,????????,???????????
  .003??????????????,??????98??????????,??2???????
  ???????????????,????99???????,??1??????????????
• ??????????,????????,????????????????

93
Bayes Theorem????
??

• (??)
• ???100,000??????????
• ??????(prior information),????100,000?????300?????
  (.3),99,700?????
• 300???????,?294?(98)???????,6????????
• ?99,700????,98,703(99)?????,997???????????

94
Bayes Theorem????
??

• (??)

P(???????????) 294/1291 .2277
95
Bayes Theorem????
??

• (??)A1???? A2????
• D1???????? D2????
• ?? P(D1).003 ? P(D2).997
• P(A1?D1).98
• P(A2?D2).99 ? P (A1?D2).01

96
Tree Diagrams
??
A1?
.98
(.00294)/(.00294.00997) .2277
D1??
.02
A2?
.003
A1?
.01
.997
D2??
.99
A2?
97
Bayes Theorem????
??

• E1 ??1??????
• E2 ??2??????
• A ?????
• ?? P(AE1).02 P(AE2).03
• P(E1) .40 P(E2) .60
• ????????????,?????????1??????

98
Tree Diagrams
??
A
.02
(.008)/(.008.018) .308
E1
.98
Ac
.4
A
.6
.03
E2
.97
Ac
99
??

• ????????????????????27,26,24,23,??????????????
  ??????????10, 25, 30, 35??????????????????????
  ?,????,?????????????????(??)
• P(???) .27 .9 .26 .75 .24 .70 .23 .65
  .7555
• P(??????) (.23 .65)/.7555 .1979

100
Fundamental rule of counting
??

• ???A?n1???
• ??B?n2???
• ???A???B?????????
• (n1 n2) ?
• ????A ???
• ??B ?????
• ??????????2 3 6?

101
Factorial Notation
??

• ?N ?????.
• The product of all integers from 1 to N is called
  N factorial and is denoted N!
• N!N(N-1)(N-2)(3)(2)(1)
• We define 0! 1

102
Permutation??
??

• A permutation of N different things taken R at a
  time, denoted NPR or PN, R is an arrangement in a
  specific order of any R of the N things.

103
Permutation??
??

• NP3

N
N-1
N-2
n1
n2
n3
104
Permutation??
105
Permutation??
??

• ????10????,??????,?????????????,???????????

106
Combination??

• A combination of N things taken R at a time,
  denoted NCR, is an arrangement of any R of these
  things without regard to order.
• N????R?,????R??????
• ????????????????????
• ???????24???,??9?????,???????
• ???????12???,???5?????,???????

107
Combination??

• ?A, B, C, D?????P4,2 4 3

???????? ???????????
108
Combination??

• ?A, B, C, D?????P4,3 4 3 2

3????3! 3 2 6 ?????,?6??????????????
109
Combination??
R????R!?????
110
Number of possible sample

• The number of possible samples of size n from a
  population of size N is CN, n
• ????????50??????5??????????,?????????????
• C50, 52,118,760

111
EXCEL function

• PN, R
• PERMUT(N, R)
• NCR
• COMBIN(N, R)