3TQ3_2024_Lecture1.pdf

ELEC ENG 3TQ3 LECTTURE 1 SEP 3RD  OUTLINE COURSE OUTLINE – AVENUE COMMUNICATION – MS Teams preferred, IF EMAIL put 3TQ3 in Subject line AVENUE will be used for announcements, outlines, lecture notes, HW assignments LECTURES will...

ELEC ENG 3TQ3 LECTTURE 1 SEP 3RD  OUTLINE COURSE OUTLINE – AVENUE COMMUNICATION – MS Teams preferred, IF EMAIL put 3TQ3 in Subject line AVENUE will be used for announcements, outlines, lecture notes, HW assignments LECTURES will be recorded but TUTORIALS will not, the problems will be posted though GRADING COMPONENTS : midterm and final – choose 3 out of 6 to ensure section consistency, assignments (4) are different. Office hours: Thursday afternoon 2:00 to 3:00 p.m. or by appointment TA names and office hours will be posted later  INTRODUCTION The theory of probability origins are closely associated with gambling It is a convenient way to introduce basic notions Understanding expected value, variance, probability of event are tools Expressing real world problem in probabilistic terms is what really matters Compare it to finding an area of the surface using integral  EXAMPLE Calculate probability that a hair dryer will work longer than 2 years given the exponential distribution of the lifetime Given historic data of lung cancer patient GUESS/ESTIMATE probability that a particular patient will live longer than 6 months Probability – given model (distribution will get back to this later) find important information Statistics – given real world data guess model and then use probabilistic tools to find information of interest  BUT…WHY ? Would you take bulky umbrella in your hand if probability of rain is 0.01? How about 0.1? How about 0.5? How about 0.9999? Answers depend on many things More important: probability of N neonates having epileptic seizures during night shift should be taken into account when deciding on the NICU budget  BASIC NOTION Toss a coin : head (H) or tail (T) Roll a six-sided die: numbers 1-6 Cards in the deck 52=4 x 13 Fair coin, fair dice, shuffled deck – equally likely outcomes (we will get back to this) Experiment and corresponding sample space (set)  QUESTION SAMPLES Distribute all cards to 4 players (13 per player). What is the probability that each player gets exactly one ace? 134 Ans: 52 (we will do this problem in tutorial next week). 4 We will revisit the notion of ratio of favorable outcomes vs all possible outcomes It all began with a letter  CASE 0 Letter by Chevalier de Mere to Blaise Pascal I used to bet even money that I would get at least one 6 in four rolls of a fair die. The probability of this is 4 times the probability of getting a 6 in a single die, i.e., 4/6=2/3; clearly I had an advantage and indeed I was making money. Now I bet even money that within 24 rolls of two dice I get at least one double 6. This has the same advantage (24/36 = 2/3), but now I am losing money. Why? A 3TQ3 student: But wait it is not 4/6 ;) – not exactly that ; correspondence between Pascal and Fermat.  EQUALLY LIKELY OUTCOMES In this course we state this for you almost always In real world you need to think carefully if this is true Perform experiment Set of all the possible outcomes S Event E consists of several outcomes Total number of “good” outcomes m, total number of outcomes n Probability of E is m/n What does that mean?  EXAMPLE Probability of even die is 0.5? What does that mean? Question is what will you do with that information? Why do you need it ?  SET THEORY To formalize the language and mathematical formulations we use set theory We know by set we refer to collection of things Set of engineering students, set of red cars in Hamilton, set of patients with hypertension in HHSC By 𝑥 ∈ ℚ we mean x belongs to set Q Note you are quite familiar with this : 2.5 ∈ ℛ but 1 + 𝑖 ∉ ℛ  SET OPERATORS UNION : 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 ⟺ 𝑥 ∈ (𝐴 ∪ 𝐵) INTERSECTION: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ (𝐴 ∩ 𝐵) COMPLEMENT: 𝑥 ∈ ℚ𝑐 ⇔ 𝑥 ∉ ℚ VENN DIAGRAM A B  EXAMPLE Gino’s offers two types of pizza: Neapolitan crust (N) or Tuscan crust (T). In addition, each pizza may have mushrooms or onions as indicated in the photo below