1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch.9 Probability 1.Chances, Probabilities 2.Marbles in Boxes: drawing with or without replacement 3.Listing the Ways 4.Venn Diagram & Exclusive Events 5.The Addition Rule 6.Conditional Probabilities 7.The Multiplication Rule 8.Partition & Bayes ’ Theorem 9.Independence 10.Exclusiveness & Independence, Addition & Multiplication

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/33 INDEX Marbles in Boxes : drawing with or without replacement 1 Chances, Probabilities 2 3 Listing the Ways 4 Venn Diagram & Exclusive Events 5 The Addition Rule

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/33  Frequentist View The Chances of something gives the percentage of time it is expected to happen, when the basic process is done over and over again, independently and under the same conditions.  Subjective View Typically in cases where repeated trial is impossible. Subjective representation of how likely it is. Defined regardless of repetition. Probabilities? 1. Chances, Probabilities

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/33  Probabilities are between 0% and 100%.  If the probability that an event A will occur is P(A), the probability that the event A will not occur is P(A). “ The event A will not occur ” is also considered another event, which is called “ complementary event A C of A ” Properties of Probability 1. Chances, Probabilities

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/33 INDEX Marbles in Boxes : drawing with or without replacement 1 Chances, Probabilities 2 3 Listing the Ways 4 5 Venn Diagram & Exclusive Events 5 The Addition Rule

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/33 Marbles in Boxes  The probability of drawing red marbles from A or B? Box A (red marbles 3, blue marble 2) Box B (red marbles 30, blue marble 20) 2. Marbles in Boxes : drawing with or without replacement

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/33 Drawing with or without replacement 12 3 12123 drawing WITH replacement 2 nd trial 1 st trial Suppose that the outcome of the 1 st trial is card 3 The outcome of the 2 nd trial is depend upon whether drawing with or without replacement 2. Marbles in Boxes : drawing with or without replacement Drawing WITHOUT replacement

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/33 INDEX 2 341 Chances, Probabilities Marbles in Boxes : drawing with or without replacement Listing the Ways Venn Diagram & Exclusive Events 5 The Addition Rule

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/33 3. Listing the Ways  Throwing a Pair of Dice Possible Ways

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/33  Combinations with a total of 9,10 rolled with three dice 9 : (1,2,6), (1,3,5), (1,4,4), (2,5,5), (2,3,4), (3,3,3) 9 : (1,2,6), (1,3,5), (1,4,4), (2,5,5), (2,3,4), (3,3,3) 10: (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 10: (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) Same in number of combinations → Same in total possible ways? No! Total of 9# of combs.Total of 10# of combs. 1,2,661,3,66 1,3,561,4,56 1,4,432,2,63 2,2,532,3,56 2,3,462,4,43 3,3,313,3,43 Total 25 27 3. Listing the Ways

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/33 INDEX 2 341 Chances, Probabilities Marbles in Boxes : drawing with or without replacement Listing the Ways Venn Diagram & Exclusive Events 5 The Addition Rule

12. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/33 4. Venn Diagram & Exclusive Events Venn Diagram  Venn Diagram A Venn diagram is a diagram using a rectangle and some inner circles to represent one or more events ABAB

13. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/33  If some two events cannot come together, the two events are called ‘ Exclusive Events ’ or ‘ Mutually Exclusive. ’ Disjoint Events ABAB 1,3,5652 1 3 (a) Exclusive Events(b) Not Exclusive 4. Venn Diagram & Exclusive Events

14. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/33 INDEX 2 341 Chances, Probabilities Marbles in Boxes : drawing with or without replacement Listing the Ways Venn Diagram & Exclusive Events 5 The Addition Rule

15. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/33 5. The Addition Rule  P(A or B) : the Probability that at least one event will occur among the two  P(A and B): the probability that the two events will come together If they are mutually exclusive, the Probability is 0.  Generalized Addition Rule: P(A or B)=P(A) + P(B) - P(A and B) Addition Rule

16. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/33 INDEX The Multiplication Rule 6 Conditional Probabilities 7 8 Partition & Bayes ’ Theorem 9 Independence 10 Exclusiveness & Independence, Addition & Multiplication

17. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/33 6. Conditional Probabilities Ex) A deck of cards is shuffled and the top two cards are put on a table, face down. You win \1,000 if the second card is Q of hearts. a) What is the probability of winning the won? b) You turn over the first card. It is the seven of clubs. Now what is the probability of winning? Conditional Probability a)Non-conditional probability  Pr(the 2 nd card is Q of hearts) ☞ 1/52 b) Conditional probability  Pr ( 2 nd card is Q of hearts | 1 st card is 7 of clubs ) ☞ 1/51

18. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/33 INDEX The Multiplication Rule 6 Conditional Probabilities 7 8 Partition & Bayes ’ Theorem 9 Independence 10 Exclusiveness & Independence, Addition & Multiplication

19. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/33 7. The Multiplication Rule  Joint Probability P(A and B), the probability the two will come together  Conditional Probability P(A|B), the probability that event A will occur given the occurrence of event B  Mmarginal Probability P(A) or P(B), non-conditional probability Multiplication Rule

20. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/33  Narrow Meaning: When some two events are mutually independent, the probability that the two will come together is acquired by multiplying each non-conditional probability. P(A and B)=P(A)·P(B)  Generalized Multiplication Rule: The probability that both of two events will occur is acquired by multiplying the probability of one event ’ s occurrence and the conditional probability of another event ’ s occurrence given the occurrence of the event. P(A and B)=P(A)·P(B|A) 7. The Multiplication Rule Multiplication Rule

21. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/33 INDEX The Multiplication Rule 6 Conditional Probability 7 8 Partition & Bayes ’ Therem 9 Independence 10 Exclusiveness & Independence, Addition & Multiplication

22. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/33 8. Partition & Bayes ’ Theorem  Partition A division of a set into Collectively Exhaustive and Mutually Exclusive events Ex) when a die is rolled, the event of even numbers and the event of odd numbers make up a partition Counter Ex) Event of odd numbers and Event of 6. Event of odd numbers and Event of numbers larger than 2. Concept

23. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/33 Partition of Union & Partition of B SAACAC A B A C and B A and B =+ =+  Partition of Union  Partition of B 8. Partition & Bayes ’ Theorem

24. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 24/33  Conditional Probability P(A|B) Probability that event A will occur given the occurrence of event B. Relative magnitude of event ( A & B ) compared with event B Circle of Right Side the Convex Lens = + == Conditional Probability 8. Partition & Bayes ’ Theorem

25. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 25/33 Two routes to event B: collectively exhaustive & mutually exclusive Tree Diagram A ACAC B given A B given A C 8. Partition & Bayes ’ Theorem

26. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 26/33 Simple form : if P ( B ) > 0 Q) If one selected the right answer to the multiple choice question having 4 possible answers ( event B), The probability that one selected it knowing surely ( event A)?  Prior Probability : P (A)=1/2  Posterior Probability : P (A|B)=4/5 Bayes Theorem (1) 8. Partition & Bayes ’ Theorem

27. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 27/33 8. Partition and Bayes ’ Theorem an example of partition

28. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 28/33 Generalized form if P ( B ) > 0, if P ( B ) > 0, Let A   , A form a partition for S Let A 1, A 2,   , A m form a partition for S Bayes ’ Theorem (2) 8. Partition and Bayes ’ Theorem

29. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 29/33 INDEX 6 Conditional Probability 7 8 Partition & Bayes ’ Therem 9 Independence 10 The Multiplication Rule Exclusiveness & Independence, Addition & Multiplication

30. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 30/33 9. Independence  If the PROBABILITY that the other event occur is not changed whether one event occur or not, we call the two events are ‘ independent ’. Otherwise, we call them ‘ dependent ’  If event A and event B are independent, P(A|B)=P(A) P(B|A)=P(B)  Narrow Meaning of Multiplication Rule : P(A and B) = P(A)  P(B) Independence & Dependence

31. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 31/33 INDEX 6 Conditional Probability 7 8 Partition & Bayes ’ Therem 9 Independence 10 The Multiplication Rule Exclusiveness & Independence, Addition & Multiplication

32. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 32/33 10. Exclusiveness & Independence, Addition & Multiplication  Mutual Exclusiveness if one event occurs then the other cannot occur  Mutual Independence If the probability that the other event occur is not changed whether one event occur or not  Mutually Exclusive events are Mutually Dependent If events A, B are mutually exclusive and event A has occurred, the probability that event B occurs becomes 0 Mutual Exclusiveness & Mutual Independence

33. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 33/33  Addition Rule Regarding the probability that at least one event will occur, Addition rule of narrow meaning is possible only when the events are mutually exclusive. (otherwise, one should subtract the overlapped part)  Multiplication Rule Regarding the probability that the two events come together, Multiplication rule of narrow meaning is possible only when the events are mutually independent. (otherwise, one should multiply the marginal probability of one event and the conditional probability of the other event.) Addition Rule & Multiplication Rule 10. Exclusiveness & Independence, Addition & Multiplication