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PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 5 PROBABILITY CORRECTIONS 1. Central tendency: Mean, Median & Mode nominal There are a few extreme scores in the distribution Some scores have undetermined values There are open ended distribution The data measured on an ordinal scale CORRECTIONS MISCONCEPTION: Many of you wrote: “for the sample A, the semiinterquartile range is more appropriate, because the semi-interquartile of sample A is smaller than that of sample B.” Two samples are as follows: Sample A: 7, 9, 10, 8, 9, 12 Sample B: 13, 5, 9, 1, 17, 9 CORRECTIONS 2. Variability: Range, Semi-interquartile range, variance, standard deviation 1.Extreme scores. 2. Sample size. 3. Stability under sampling 4. Open-ended distributions CORRECTIONS Calculating sample standard deviation: Population Sample Mean µ X variance σ2 = SS/N s2=SS/n-1 Standard deviation σ = √SS/N s = √SS/n-1 Interpretations of Probability 1. The frequency interpretation of probability The probability that some specific outcome of a process will be obtained can be interpreted to mean the relative frequency with which that outcome would be obtained if the process were repeated a large number of times under similar conditions. Interpretations of Probability 2. The classical interpretation of probability It is based on the concept of equally likely outcomes. Interpretations of Probability 3. The subjective interpretation of probability The probability that a person assigns to a possible outcome of some process represents her/his own judgment of the likelihood that the outcome will be obtained. This judgment will be based on each person’s beliefs or information about the process. It is appropriate to speak of a certian person’s subjective probability , rather than to speak of the true probability of that outcome. Experiments An experiment is the process of making observation. Ex: a. A coin is tossed 10 times. The experimenter might want to determine the probability that at least four heads will be obtained. b. In an experiment in which a sample of 1000 transistors is to be selected from a large shipment of similar items and each selected item is to be inspected, a person might want to determine the probability that not more than one of the selected transistors will be defective. Sample space A sample space is a set of points corresponding to all distinctly possible outcomes of an experiment. Ex: For the die tossing experiment, Sample point A sample point is a point in a sample space. Ex: For the die tossing experiment, Descrete sample A descrete sample space is one that contains a finite number or countable infinity of sample points. Ex: A coin is tossed two times. Event For a descrete sample space, an event is any subset of it. Ex: a. A coin is tossed two times. b. For the die tossing experiment Note: simple event Ex: observe a 6. Summarizing example Tossing a Coin: Suppose that a coin is tossed three times. Then Experiment : Sample space : Sample point: Events: Simple event: Definition of probability Axiom 1. For every event A, Pr (A)≥0. Axiom 2. Pr (S) = 1. Axiom 3 For every infinite sequence of disjoint events Ai , A2 ,..... Pr Ai Pr( Ai ) i 1 i 1 Theorem 1 Pr 0 Theorem 2 For every finite sequence of n disjoint events Ai , A2 ,..... n n Pr Ai Pr( Ai ) i 1 i 1 Theorem 3 For every event A Pr( A) 1 Pr( A) Theorem 4 If A B , then Pr( A) Pr( B) Theorem 5 For every event A, 0 Pr( A) 1 Theorem 6 For every two events A and B, Pr( A B) Pr( A) Pr( B) Pr( A B) Summarizing example Diagnosing Diseases: A patient arrives at a doctor’s office with a sore throat and low grade fever. After an exam, the doctor decides that the patient has either a bacterial infection or a viral infection or both. The doctor decides that there is a probability of 0.7 that the patient has a bacterial infection and a probability of 0.4 that the person has a viral infection. What is the probability that the patient has both infection? Summarizing example 2 Demands for Utilities: A contractor is building an office complex and needs to plan for water and electricity demands (sizes of pipes, conduit, and wires). After consulting with prospective tenants and examining historical data, the contractor decides that the demand for electricity will range between 1 million and 150 million kilowatt-hours per day and water demand will be between 4 and 200 (in thousand gallons per day). All combinations of ellectrical and water demand are considered possible. Finite sample space Experiments include a finite number possible outcomes. of S s1 , s2 ,......., sn The number pi is the probability that the outcome of the experiment will be si , (i 1, 2,3,...., n) pi 0 n p i 1 i 1 If the probability assigned to each of the outcomes is 1/n, then this sample space S is a simple sample space. Summarizing example Fiber breaks : consider an experiment in which five fibers having different lenghts are subjected to a testing process to learn which fiber will break first. Suppose that the lenghts of the five fibers are 1, 2, 3, 4, and 5 meters, respectively. Suppose also that probability that any given fiber will be the first to break is proportional to the lenght of that fiber. Determine the probability that the lenght of the fiber that breaks first is not more than 3 meters. The probability of a union of events If the events are disjoint, n n Pr Ai Pr( Ai ) i 1 i 1 Theorem: For every three events, Pr( A1 A 2 A3 ) ... Summarizing example Student Enrollment: Among a group of 200 students, 137 students are enrolled in a mathemtical class, 50 students are enrolled in a history class, and 124 students are enrolled in a music class. Furthermore, the number of students enrolled in both the mathematics and history classes is 33; the number enrolled in both the history and music class 29, and the number enrolled in both the methemtics and music class is 92. Finally, the number of students enrolled in all three classes is 18. Determine the probability that a student slected at random from the group of 200 stundents will be enrolled in at least one of the three classes. Teaching probability Constructing probability examples Work with examples such as the probability of boy and girl births and use probability models of real outcomes. These are more interesting and are known than card and crap games. Teaching probability Random numbers via dice or handouts a. Rolling the dice ones gives a random digit. If it is too inconvenient, you can prepare handouts of random numbers for your students. c. You can use already existing material. Ex: telephone book. b. Teaching probability Probability of compound events Use “babies” or “real vs. fake coin flips”. Babies: Students enjoy examples involving families and babies. EX: We adapt a standard problem in probability by asking students which of the following sequences of boy and girl births is most likely, given that a family has four children: bbbb, bgbg, or gggg. Teaching probability Probability of compound events Real vs. fake coin flips: Students often have diffuculties with probability of distributions. We pick two students to be “judges” and one to be the “recorder” and divide the others in the class into two groups. One group is instructed to flip a coin 100 times, or flip 10 coins 10 times each, or follow some similarly defined protocol, and then to write the results, in order, on a sheet of paper, writing heads as “1” and tails as “0”. The second group is instructed to create a sequence of 100 “0”s ans “1”s that are intended to look like the result of coin flips- but they are to do this without flipping any coins or randomization device- and to write this sequence on a sheet of paper. Teaching probability Probability modeling We can apply probabilty distributions to real phenomena. Ex: Airplane failure (and other rare events) Looking back historical data gave probability estimate of about 2%. Its deadly accident was calculated as 82%. what is the probabilty of that a person will be dead in an airplane accident due to airplane failure?