Elementary Probability. Definition Three Types of Probability Set operations and Venn Diagrams ...

Elementary Probability

Definition Three Types of Probability Set operations and Venn Diagrams Mutually Exclusive, Independent and

Dependent Events (Rule of Addition, Rule of Multiplication,

Conditional Probability)

Probability The study of how likely it is that an event will occur.

compare trial experiment event sample space

Probability Sample space

Toss a coin twice and observe the possible outcomes.

Toss a coin twice and observe times a head appears.

S1 = { (HH), (TT), (HT), (TH) }

S2= {0, 1, 2}

Probability

)()(

)(SnEn

EP

P(E): probability of an event, E, occurring.n(E): number of ways the event can occur.n(S): total number of outcomes =sample space

(0~1)

ProbabilityOne card is selected from a pack of 10 cards numbered 1 to 10. Sample space: 1, 2, 3 ,4, 5, 6, 7, 8, 9, 10 (10)Calculate the probability of:a) Selecting a 5b) Selecting an odd cardc) Selecting a card less than 5

ProbabilityOne card is selected from a pack of 10 cards numbered 1 to 10.

)()(

)(SnEn

EP 101

Calculate the probability of:a) Selecting a 5

n(E): 5 (1)

ProbabilityOne card is selected from a pack of 10 cards numbered 1 to 10.

Calculate the probability of:b) Selecting an odd card

)()(

)(SnEn

EP 105

21

n(E): 1, 3, 5, 7, 9 (5)

ProbabilityOne card is selected from a pack of 10 cards numbered 1 to 10.

Calculate the probability of:c) Selecting a card less than 5

)()(

)(SnEn

EP 104

52

n(E): 1, 2, 3, 4 (4)

Three Types of Probability Classical Probability a. finite b. equal possibilityRelative Frequency ProbabilitySubjective Probability

Set Operations and Venn Diagram Set theory forms the basis for probability

applications. A set is a collection of objects or elements. Elements are shown inside parentheses {}

e.g.Draw a card from a pack numbered 1 to 5S = {1,2,3,4,5}

Set Operations and Venn DiagramSubset refers to some of the elements of S. Draw a card from a pack numbered 1 to 5S = {1,2,3,4,5}Subset:{1,2}, {3,4}, {2,3,5}, etc.

1

3

Venn DiagramS

A

SA

45 6 2

Set Operations

BA

A = {1, 2, 3, 5 } B = {1, 2, 4, 5 }

= {1, 2, 3, 4, 5}

BA = {1,2,5}

Set Operations

A

)(1)( APAP

A

Mutually ExclusiveTwo or more events are mutually exclusive if the occurrence of any one of them excludes the occurrence of all the others. That is, only one can happen.

P(A or B) = P(A) + P(B) – P(A and B)

Rule of Addition

P(A or B) = P(A) + P(B)

P(A+B) P(A B) ∪

Mutually Exclusive

P(A or B) = P(A) + P(B)

P(A or B) = P(A) + P(B) – P(A and B)

A B

Two magazines:Magazine A 26%Magazine B 18%

What is the probability for people who read one of the magazines?

P(A+B) = P(A) + P(B) =26% + 18%=44%

Mutually ExclusiveP(A or B) = P(A) + P(B) – P(A and B)

BA

Two magazines:Magazine A 26%Magazine B 18%Both magazine 5%

What is the probability for people who read at least one of the magazines?

P(A+B) = P(A) + P(B) - P(AB) =26% + 18% - 5% = 39%

32

9015

9030

9045

Think Pick a number from 10 to 99 A: {The number can be divided by 2} B: {The number can be divided by 3}

What is the probability for picking a number which can be divided by 2 or 3?

P(A+B) = P(A) + P(B) - P(AB) =

Independent EventsTwo or more events are said to be independent if the occurrence or non-occurrence of one of them in no way affects the occurrence or non-occurrence of the others. The events are unconnected.

P(A and B) = P(A) × P(B)

Rule of Multiplication

P(AB) P(A∩B) P(A×B)

Independent events Throw a coin and a dice at the same time.Calculate:The probability of a head and a 5 at the same time.A: { Get a head at random}B: { Get a 5 at random}

)()()( BPAPBAP121

61

21

Dependent Events Conditional Probability

Two or more events are said to be dependent when the probability that event B takes place is subject to whether event A has taken place. In other words, the prior occurrence of event A affects the probability of event B occurring.

Dependent Events Conditional Probability

10 products5 nonconforming products3 inferior products2 waste products

Calculate: a) the probability of selecting a waste product.b) The probability of selecting a waste product

given that a nonconforming product is selected.

Dependent Events Conditional Probability

10 products5 nonconforming products3 inferior products2 waste products

A= { Select a waste product }B= { Select a nonconforming product }

Dependent Events Conditional Probability

10 products5 nonconforming products3 inferior products2 waste products

Calculate: a) the probability of selecting a waste product.

)()(

)(SnAn

AP 51

102

Dependent Events Conditional Probability

10 products5 nonconforming products3 inferior products2 waste products

A= { Select a waste product }B= { Select a nonconforming product }

Conditional Probability10 products5 nonconforming products3 inferior products2 waste products

Calculate: b) The probability of selecting a waste product

given that a nonconforming product is selected.

52

P(A|B))()(

BnABn

)(

)()(

)(

SnBn

SnABn

)()(

BPABP

Dependent Events Conditional Probability

P(A | B) =)()(

BPABP

P(A B) = P(B) P(A|B)

more

What about P(ABC)?

Sampling Inference Estimation Point Estimation Interval Estimation

Hypothesis Testing

Normal Distribution

Normal Distribution The distribution of many common variables

such as height, weight, shoe-size and life-expectancy approach what is known as a normal probability distribution.

Feature 1. The mean, median and mode are equal and are

at the centre of the distribution.

2. A normal distribution is symmetrical about the mean. (bell-shaped)

4. The area under the whole graph =1, so the area under half the graph=0.5

3. The probability equals the area under the graph.

Feature

95.5%

99.7%

68%

Calculations involving ND

xz

X is the value under considerationμ is the population meanσ is the population standard deviationZ the number of standard deviations the value is away from the mean.

115 100 115

xz

What percentage of people have an I.Q. between 115 and 140?

Average I.Q. =100

100

100 , 15

115

When x=115

140

What percentage of people have an I.Q. between 100 and 115?

Average I.Q. =100

100

100 , 15

115

What percentage of people have an I.Q. between 85 and 115?

Average I.Q. =100

100

100 , 15

11585

What percentage of people have an I.Q. over 120?

Average I.Q. =100

100

100 , 15

120

What percentage of people have an I.Q. less than 85?

Average I.Q. =100

100

100 , 15

85

What percentage of people have an I.Q. less than 135?

Average I.Q. =100

100

100 , 15

135

What percentage of people have an I.Q. over 119?

Average I.Q. =100

100

100 , 15

119

What I.Q. would you need to have in order to be in the top 10% of I.Q.s?

Average I.Q. =100

100

100 , 15

x

10%

In formulating a budget, the value of sales is expected to be $1.2 million, with a standard deviation of $200,000.

Within what range can management be 90% confident that sales will fall?

1.2 , 0.2

1.2 x2x1

45%45%

Central Limit Theorem If sufficient samples are randomly drawn from

a population, then the distribution of the sample mean will be normally distributed about the population mean.

Central Limit Theorem Calculation of probability when a sample size

is given Calculation of the mean

Central Limit Theorem Calculation of probability when a sample size

is given

840 850 10 2 0.6715 15 3

xz

A company manufacturing drinking straws has calculated that the mean contents of a carton is 850 with a standard deviation of 15.

What is the probability that a carton will contain under 840?

850

850 , 15

840When x=840

z-table, reading of 0.67 = 0.24857

P(<840)=0.5-0.24857=0.25143=25.13%

15850840

xz

A company manufacturing drinking straws has calculated that the mean contents of a carton is 850 with a standard deviation of 15

What is the probability that a carton will contain under 840?

850

850 , 15

840When x=840

, a sample of 9 cartons was taken.

A company manufacturing drinking straws has calculated that the mean contents of a carton is 850 with a standard deviation of 15, a sample of 9 cartons was taken.

What is the probability that a carton will contain under 840?

59

15

nx

25

850840

xz

P(<840)=0.5-0.47725=0.0228=2.28%

Central Limit Theorem Calculation of the mean

x f fx0 1 01 3 32 4 83 2 6

10 17

17 1.710

fxf

x f fx p0 1 0 0.11 3 3 0.32 4 8 0.43 2 6 0.2

10 17 1.0

7.1)2.03()4.02()3.01()1.00()( xE

x f0 1 2.89 2.891 3 0.49 1.472 4 0.09 0.363 2 1.69 3.38

10 8.10

7.1x

2)( xxf

f

xxf 2)( 90.0

1010.8

2( )x x

x p(f)0 0.1 0.2891 0.3 0.1472 0.4 0.0363 0.2 0.338

1 0.81

7.1x

2)( xxf

f

xxf 2)( 90.0

181.0

Hypothesis Testing

Sampling Inference Estimation Point Estimation Interval Estimation

Hypothesis Testing

Area of Rejection Area of Non-rejection

%98X %85x

H0: 98%H1: < 98%

Area of RejectionArea of Non-rejection

%3X %7x

H0: ≤ 3%H1: > 3%

Area of RejectionArea of Non-rejection

gX 200 gx 180

H0: = 200gH1: > 200g or <200g

Area of Rejection

Hypothesis Testing Law of Large Numbers Small Probability Events

2

21

Hypothesis Testing Law of Large Numbers Small Probability Events

Area of Rejection Area of RejectionArea of Non-rejection

Steps1. Determine the null and alternative hypotheses2. Determine the level of significance3. Determine test statistic (z or t)4. Determine the critical value5. Calculate the value of the test statistic6. Make decisions to accept or reject the null

hypothesis.

A woman is considering buying a business. The owner of the business, a delicatessen, claims that the daily turnover follows an approximate normal curve with an average of $580 and a standard deviation of $50. The potential investor samples the takings over 30 days and calculates the average takings as $550.

Use a significance level of 0.01 to determine if the claim of the present owner of the delicatessen is valid or not.

Step 1 Determine the null and alternative hypotheses H0: $580 H1: < $580

Step 2 Determine the significance level level of significance 0.01

Step 3 Determine test statistic z: large sample (n 30) a sample from a normal distribution

t: a small sample that is not from a normal

distribution when the value of the standard deviation must be

estimated

Step 3 Determine test statistic normal distribution standard deviation sample size

ND SD known large/small sample size: Z SD unknown large sample size : Z, s SD unknown small sample size : t, s

Non ND large sample size Z

Step 4 Determine the critical value Z= 2.33 Reject if |Z|>2.33

Area of Rejection Area of Non-rejection

Significance Level

Area of Rejection |Z|

One Tailed Test Two-Tailed Test

0.05 1.65 1.960.01 2.33 2.580.001 3.09 3.30

table

Step 4 Determine the critical value

Area of Rejection Area of RejectionArea of Non-rejection

Two tailed test

Step 4 Determine the critical value

Area of Rejection Area of Non-rejection

One tailed test

Step 4 Determine the critical value

Area of RejectionArea of Non-rejection

One tailed test

Step 4 Determine the critical value Z= 2.33 Reject if |Z|>2.33

Area of Rejection Area of Non-rejection

Step 4

|Z|=3.29

29.3

3050

580550

n

xZ

Step 5 Make decisions

Z =2.33 |Z|=3.29 |Z|> Z

Reject H0The average daily takings are less than $580.

Area of Rejection Area of Non-rejection-3.29 -2.33

Error Type I error If a hypothesis is rejected when it should be

accepted.

Type II error If a hypothesis is accepted when it should

have been rejected.

Sample Population vs. Sample Benefits: Timeliness Cost Accessibility Dynamic nature of business/market

Features A sample should be representative of the

population, that is, all the characteristics that are present in the population must also be found in the sample.

Chosen at random Unbiased

Methods Simple Random or Lottery Method Systematic Sample (array) Stratified Sample Quota Sample Cluster Sample Multi-Staged

Errors in Sampling Non-sampling errors arise from the research

mechanisms used in collecting and analysing the data

Sampling error – Qualities exhibited in the sample may not be true of the population.

Procedures in a Research Project

Objective Analysis of existing data Qualitative pilot Researching the project Analysis of data and recommendations