1. STATISTICS I COURSE INSTRUCTOR: TEHSEEN IMRAAN

2. CHAPTER 5 PROBABILITY

3. TYPES OF STATISTICS DESCRIPTIVE STATISTICS INFERENTIAL STATISTICS

4. WHAT IS PROBABILITY? A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. Probability is expressed either as a percent or as a decimal. The likelihood that any particular event will happen may assume values between 0 and 1.0. A value close to 0 indicates the event is unlikely to occur, whereas a value close to 1.0 indicates that the event is quite likely to occur. To illustrate, a value of 0.60 might express your degree of belief that tuition will be increased at your college, and 0.50 the likelihood that your first marriage will end in divorce.

5. EXPERIMENT, OUTCOME & EVENT Experiment: A process that leads to the occurrence of one and only one of several possible observations. Outcome: A particular result of an experiment. Event: A collection of one or more outcomes of an experiment.

6. APPROACHES TO PROBABILITY

7. CLASSICAL PROBABILITY A probability based on the assumption that outcomes of an experiment are equally likely. To find the probability of a particular outcome we divide the number of favorable outcomes by the total number of possible outcomes as shown in text formula [5-1].

8. CLASSICAL PROBABILITY

9. For example, you take a multiple-choice examination and have no idea which one of the choices is correct. In desperation you decide to guess the answer to each question. The four choices for each question are the outcomes. They are equally likely, but only one is correct. Thus the probability that you guess a particular answer correctly is 0.25 found by 1/ 4.

10. MUTUALLY EXCLUSIVE The occurrence of one event means that none of the other events can occur at the same time. For instance, an employee selected at random is either a male or female but cannot be both. A computer chip cannot be defective and not defective at the same time.

11. COLLECTIVELY EXHAUSTIVE If an experiment has a set of events that includes every possible outcome, then the set of events is called collectively exhaustive. At least one of the events must occur when an experiment is conducted. For example: In a die-tossing experiment every outcome will be either an even number or an odd number. Thus the set is collectively exhaustive. If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities equals 1.

12. EMPIRICAL CONCEPT Another way to define probability is based on relative frequencies. The probability of an event happening is determined by observing what fraction of the time similar events happened in the past. To find a probability using the relative frequency approach we divide the number of times the event has occurred in the past by the total number of observations. Suppose the Civil Aeronautics Board maintained records on the number of times flights arrived late at the Newark International Airport. If 54 flights in a sample of 500 were late, then, according to the relative frequency formula, the probability a particular flight is late is 0.108. Based on passed experience, the probability is 0.108 that a flight will be late.

13. EMPIRICAL CONCEPT

14. SUBJECTIVE PROBABILITY The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. Subjective probability is based on judgment, intuition, or "hunches." The likelihood that the horse, Sir Homer, will win the race at Ferry Downs today is based on the subjective view of the racetrack odds maker.

15. Some rules for assigning probabilities In the study of probability it is often necessary to combine the probabilities of events. This is accomplished through both rules of addition and rules of multiplication. There are two rules for addition, the special rule of addition and the general rule of addition.

16. Special Rule for Addition To apply the special rule of addition, the events must be mutually exclusive. The special rule of addition states that the probability of the event A or the event B occurring is equal to the probability of event A plus the probability of event B. The rule is expressed by using text formula [5-2] To apply the special rule of addition the events must be mutually exclusive. This means that when one event occurs none of the other events can occur at the same time.

17. Special Rule for Addition

18. Venn Diagram Venn diagrams, developed by English logician J. Venn, are useful for portraying events and their relationship to one another. They are constructed by enclosing a space, usually in a form of a rectangle, which represents the possible events. Two mutually exclusive events such as A and B can then be portrayed as in the following diagram by enclosing regions that do not overlap (that is, that have no common area).

19. Venn Diagram

20. The Complement Rule The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from one Complement rule: A way to determine the probability of an event occurring by subtracting the probability of an event not occurring from 1. In some situations it is more efficient to determine the probability of an event happening by determining the probability of it not happening and subtracting from 1.

21. The Complement Rule

22. General Rule of Addition When we want to find the probability that two events will both happen, we use the concept known as joint probability. Joint probability: A probability that measures the likelihood two or more events will happen concurrently. What if the events are not mutually exclusive? In that case the general rule of addition is used. The probability is computed using the text formula [5- 4].

23. General Rule of Addition

24. Where: P(A) is the probability of the event A. P(B) is the probability of the event B. P(A and B) is the probability that both events A and B occur.

25. General Rule of Addition For example, a study showed 15 percent of the work force to be unemployed, 20 percent of the work force to be minorities, and 5 percent to be both unemployed and minorities. What percent of the work force are either minorities or unemployed?

26. Solution P (unemployed or minority) = P (unemployed) + P (minority) - P (unemployed and minority) = 0.15 + 0.20 - 0.05 = 0.30

27. Venn diagram

28. Using addition rule

29. Rules of Multiplication There were two rules of addition, the general rule and the special rule. We used the general rule when the events were not mutually exclusive and the special rule when the events were mutually exclusive. We have an analogous situation with the rules of multiplication.

30. Special Rule of Multiplication We use the general rule of multiplication when the two events are not independent and the special rule of multiplication when the events A and B are independent. Two events are independent if the occurrence of one does not affect the probability of the other. A way to look at independence is to assume two events A and B occur at different times. For example, flipping a coin and getting tails is not affected by rolling a die and getting a two. The special rule of multiplication is used to combine events where the probability of the second event does not depend on the outcome of the first event.

31. Conditional Probability The probability of a particular event occurring may be altered by another event that has already occurred. Probability measures uncertainty, but the degree of uncertainty changes as new information becomes available. Symbolically, it is written P (B\A). The vertical line (\) does not mean divide; it is read "given that" as in the probability of B "given that" A already occurred.

32. Conditional Probability Rule of Independence: P(A)= P(A\B) The rule of independence restates our definition. It says that the probability of event A is equal to the conditional probability of A given B. In other words event A is unaffected by any prior occurrence of event B.

33. Special Rule of Multiplication P(A and B) = P(A) * P(B) As an example, a nuclear power plant has two independent safety systems. The probability the first will not operate properly in an emergency P (A) is 0.01, and the probability the second will not operate P (B) in an emergency is 0.02. What is the probability that in an emergency both of the safety systems will not operate? The probability both will not operate is: The probability 0.0002 is called a joint probability, which is the simultaneous occurrence of two events. It measures the likelihood that two (or more) events will happen together (jointly).

34. SPECIAL RULE OF MULTIPLICATION IN CASE OF THREE EVENTS The probability for three independent events, A, B, and C, the special rule of multiplication used to determine the probability of all three events will occur is: P(A and B and C)= P(A).P(B).P(C)

35. GENERAL RULE OF MULTIPLICATION The general rule of multiplication is used to combine events that are not independent that is, they are dependent on each other. For two events, the probability of the second event is affected by the outcome of the first event. Under these conditions, the probability of both A and B occurring is given in formula [5-7]. P(A and B)= P(A). P(B\A), where P (B\A ) is the probability of B occurring given that A has already occurred. Note that P (B\A) is a conditional probability.

36. GENERAL RULE OF MULTIPLICATION For example, among a group of twelve prisoners, four had been convicted of murder. If two of the twelve are selected for a special rehabilitation program, what is the probability that both of those selected are convicted murderers?

37. Solution Let A 1 be the first selection (a convicted murderer) and A 2 the second selection (also a convicted murderer). Then P(A 1 ) = 4/12. After the first selection, there are 11 prisoners, 3 of whom are convicted of murder, hence P(A 2 \A 1 ) = 3/11. The probability of both A 1 and A 2 happening is:

38. Solution

39. Contingency Table A table used to classify sample observations according to two or more identifiable characteristics A contingency table is a cross tabulation that simultaneously summarizes two variables of interest and their relationship. A survey of 200 school children classified each as to gender and the number of times Pepsi-Cola was purchased each month at school. Each respondent is classified according to two criteria-the number of times Pepsi was purchased and gender.

40. Contingency Table Gender Bought Pepsi BoysGirlsTotal 051015 1 2540 2 or more 8065145 Total100 200

41. Using the multiplication rule

42. Tree diagram The tree diagram is a graph that is helpful in organizing the calculations that involve several stages. Each segment in the tree is one stage of the problem. The branches are weighted probabilities.

43. Principles of Counting If the number of possible outcomes in an experiment is small, it is relatively easy to count the possible outcomes. However, sometimes the number of possible outcomes is large, and listing all the possibilities would be time consuming, tedious, and error prone. Three formulas are very useful for determining the number of possible outcomes in an experiment. They are: the multiplication formula, the permutation formula, and the combination formula.

44. Multiplication Rule Multiplication formula: If there are m ways of doing one thing, and n ways of doing another thing, there are m x n ways of doing both. In terms of formula, it is total no. of arrangements= (m).(n)

45. Permutation Rule The permutation is an arrangement of objects or things wherein order is important. That is, each time the objects or things are placed in a different order, a new permutation results. Any arrangements of r objects selected from a single group of n possible objects.

46. Permutation Formula Permutation Formula,

47. Permutation Formula Where: P is the number of permutations, or ways the objects can be arranged. n is the total number of objects. r is the number of objects selected. Note: the n! is a notation called "n factorial." For example, 4! means 4 times 3 times 2 times 1 = 4 × 3 × 2 × 1 = 24

48. Combination Rule One particular arrangement of the objects without regard to order is called a combination. The number of ways to choose r objects from a group of n possible objects without regard to order.

49. Combination Formula Combination Formula,

50. Combination Formula Where: C is the number of different combinations. n is the total number of objects. r is the number of objects to be used at one time.