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. S 1 = { (HH), (TT), (HT), (TH) } S 2 = {0, 1, 2}

Probability 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)

Probability One 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 5 b) Selecting an odd card c) Selecting a card less than 5

Probability One card is selected from a pack of 10 cards numbered 1 to 10. Calculate the probability of: a) Selecting a 5 n(E):5 (1)

Probability One card is selected from a pack of 10 cards numbered 1 to 10. Calculate the probability of: b) Selecting an odd card n(E): 1, 3, 5, 7, 9 (5)

Probability One card is selected from a pack of 10 cards numbered 1 to 10. Calculate the probability of: c) Selecting a card less than 5 n(E): 1, 2, 3, 4 (4)

Three Types of Probability  Classical Probability a. finite b. equal possibility  Relative Frequency Probability  Subjective 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 5 S = {1,2,3,4,5}

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

1 3 Venn Diagram S A

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

Set Operations A

Mutually Exclusive Two 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) 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 Exclusive P(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%

 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 Events Two 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) 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}

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 products 5 nonconforming products 3 inferior products 2 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 products 5 nonconforming products 3 inferior products 2 waste products A= { Select a waste product } B= { Select a nonconforming product }

Dependent Events Conditional Probability 10 products 5 nonconforming products 3 inferior products 2 waste products Calculate: a)the probability of selecting a waste product.

Dependent Events Conditional Probability 10 products 5 nonconforming products 3 inferior products 2 waste products A= { Select a waste product } B= { Select a nonconforming product }

Conditional Probability 10 products 5 nonconforming products 3 inferior products 2 waste products Calculate: b) The probability of selecting a waste product given that a nonconforming product is selected. P(A|B)

Dependent Events Conditional Probability P(A | B) = P(A B) = P(B) P(A|B) more What about P(ABC)?

Sampling Inference  Estimation Point Estimation Interval Estimation  Hypothesis Testing

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= The probability equals the area under the graph.

Feature

Calculations involving ND X is the value under consideration μ is the population mean σ is the population standard deviation Z the number of standard deviations the value is away from the mean.

What percentage of people have an I.Q. between 115 and 140? Average I.Q. =100    When x=115 

What percentage of people have an I.Q. between 100 and 115? Average I.Q. =100   

What percentage of people have an I.Q. between 85 and 115? Average I.Q. =100    

What percentage of people have an I.Q. over 120? Average I.Q. =100   

What percentage of people have an I.Q. less than 85? Average I.Q. =100   

What percentage of people have an I.Q. less than 135? Average I.Q. =100   

What percentage of people have an I.Q. over 119? Average I.Q. =100   

What I.Q. would you need to have in order to be in the top 10% of I.Q.s? Average I.Q. =100   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?   x2x2x2x2 x1x1x1x1 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

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?    When x=840 z-table, reading of 0.67 = P(<840)= = =25.13%

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?    When 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? P(<840)= =0.0228=2.28%

Central Limit Theorem  Calculation of the mean

xffx

xffxp

xf

xp(f)

Hypothesis Testing

Sampling Inference  Estimation Point Estimation Interval Estimation  Hypothesis Testing

Area of RejectionArea of Non-rejection H 0 :   1 :  98%

Area of Rejection Area of Non-rejection H 0 :  ≤   1 :  3%

Area of Rejection Area of Non-rejection H 0 :  =  g  1 :  200g or  <200g Area of Rejection

Hypothesis Testing  Law of Large Numbers  Small Probability Events

Hypothesis Testing  Law of Large Numbers  Small Probability Events Area of Rejection Area of Non-rejection

Steps 1. Determine the null and alternative hypotheses 2. Determine the level of significance 3. Determine test statistic (z or t) 4. Determine the critical value 5. Calculate the value of the test statistic 6. 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 H 0 :  $   1 :  $   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 RejectionArea of Non-rejection

Significance Level  Area of Rejection |Z|  One Tailed TestTwo-Tailed Test

 Step 4 Determine the critical value Area of Rejection Area of Non-rejection

 Step 4 Determine the critical value Area of RejectionArea of Non-rejection

 Step 4 Determine the critical value Area of RejectionArea of Non-rejection

 Step 4 Determine the critical value Z  = 2.33 Reject if |Z|>2.33 Area of RejectionArea of Non-rejection

 Step 4 |Z|=3.29

 Step 5 Make decisions Z  =2.33 |Z|=3.29 |Z|> Z  Reject H 0 The average daily takings are less than $580. Area of RejectionArea of Non-rejection

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