Introductory Probability

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1 Introductory Probability Discrete Probability Distributions Dr. Nguyen Department of Mathematics UK January 9, 2019

2 Agenda Syllabi and Course Websites Class Information Random Variables and Sample Spaces Distribution Functions Uniform Distribution Functions

3 Course Websites My website: Syllabus, slides (under construction) Canvas website: Grades, Backup link to my website Homework: Online homework assignments (under construction) Textbook:

4 Grade Breakdown Grades are based on: Category Weight Weekly Quizzes 15% Weekly Homework 15% Two Midterms 40% Final Exam 30%

5 Attendance I will start taking attendance on January 18 (next Friday). A student will receive an additional 20 points extra credit added to their homework score (max 120/100) at the end of the semester if they have no greater than three unexcused absences.

6 Exams Each week (usually Friday), there will be an in-class quiz. Exceptions include this week, and weeks with midterms. The dates for the two midterms and the nal are in the syllabus.

7 Homework Assignments You will be given problem sets at the WebWork site. Most homework problems will let you try again until you get it. Most problems are randomized. Submit each problem using a button on each page.

8 Reading Assignments I will give suggestions for reading in preparation for each lecture. Please read the recommended sections before lecture.

9 Coin Tosses We can use a computer to simulate many tosses of a coin. Note: You may want to explore the textbook website link in the syllabus for programs that perform these simulations if you want to try them at home.

10 @Home: Coin Toss Results When I ran the coin toss simulations, I got results like these: I once got only one head in ten tosses! Another time, I got 6 heads in 10 tosses. Sometimes, I got 5 heads in 10 tosses. Since we expect heads to come up half of the time, we should expect 5 heads in 10 tosses.

11 @Home: Coin Toss Results When I ran the coin toss simulations with larger numbers of tosses, With 100 tosses, I usually got between heads in dierent simulations. We would expect 50 heads. With 1,000 tosses, I usually got between heads. We would expect 500 heads. With 10,000 tosses, I usually got between 4,950-5,050 heads. We would expect 5,000 heads.

12 Frequency Concept of Probability If an experiment has outcome (result) A with probability 0 p 1, then if we repeat the experiment many times, we should expect that the fraction of times that outcome A occurs is p.

13 Frequency Concept of Probability If an experiment has outcome (result) A with probability 0 p 1, then if we repeat the experiment many times, we should expect that the fraction of times that outcome A occurs is p. What is many? What is expect?

14 Random Variables and Sample Spaces (Def 1.1) Suppose we have a chance experiment. A random variable (like X ) represents the outcome. The sample space Ω is the set of all possible outcomes. Elements of a sample space are outcomes. Subsets of a sample space are events. A sample space is discrete if it is nite or countably innite. Note: outcomes are denoted by lowercase letters and events by uppercase letters.

15 Random Variables and Sample Spaces (Def 1.1) Suppose we have a chance experiment. A random variable (like X ) represents the outcome. The sample space Ω is the set of all possible outcomes. Elements of a sample space are outcomes. Subsets of a sample space are events. A sample space is discrete if it is nite or countably innite. Note: outcomes are denoted by lowercase letters and events by uppercase letters.

16 Random Variables and Sample Spaces (Def 1.1) Suppose we have a chance experiment. A random variable (like X ) represents the outcome. The sample space Ω is the set of all possible outcomes. Elements of a sample space are outcomes. Subsets of a sample space are events. A sample space is discrete if it is nite or countably innite. Note: outcomes are denoted by lowercase letters and events by uppercase letters.

17 Random Variables and Sample Spaces (Def 1.1) Suppose we have a chance experiment. A random variable (like X ) represents the outcome. The sample space Ω is the set of all possible outcomes. Elements of a sample space are outcomes. Subsets of a sample space are events. A sample space is discrete if it is nite or countably innite. Note: outcomes are denoted by lowercase letters and events by uppercase letters.

18 Answering a Phone A family (dad, mom, kid) has a phone. When the phone rings, one of them picks it up. We can let X record who picked up the phone. The sample space is all the family members: Ω = {dad, mom, kid}. Each person is an outcome. An example of an event E is A parent picks up the phone. This event has two outcomes: E = {dad, mom}. Another event is F : The Kid picks up the phone. This event has one outcome: F = {kid}.

19 Answering a Phone A family (dad, mom, kid) has a phone. When the phone rings, one of them picks it up. We can let X record who picked up the phone. The sample space is all the family members: Ω = {dad, mom, kid}. Each person is an outcome. An example of an event E is A parent picks up the phone. This event has two outcomes: E = {dad, mom}. Another event is F : The Kid picks up the phone. This event has one outcome: F = {kid}.

20 Answering a Phone A family (dad, mom, kid) has a phone. When the phone rings, one of them picks it up. We can let X record who picked up the phone. The sample space is all the family members: Ω = {dad, mom, kid}. Each person is an outcome. An example of an event E is A parent picks up the phone. This event has two outcomes: E = {dad, mom}. Another event is F : The Kid picks up the phone. This event has one outcome: F = {kid}.

21 Answering a Phone A family (dad, mom, kid) has a phone. When the phone rings, one of them picks it up. We can let X record who picked up the phone. The sample space is all the family members: Ω = {dad, mom, kid}. Each person is an outcome. An example of an event E is A parent picks up the phone. This event has two outcomes: E = {dad, mom}. Another event is F : The kid picks up the phone. This event has one outcome: F = {kid}.

22 Distribution Functions (Def 1.2) Suppose our experiment has a nite number of outcomes. Let X be the random variable and Ω be the sample space. A distribution function for X is a real-valued function m whose domain is Ω and which satises: 1. m(ω) 0 for all outcomes ω Ω. 2. m(ω) = 1. ω Ω For any event E, the probability of E is P(E) = m(ω). ω E In particular, for the event with just one outcome {ω}, P({ω}) = m(ω).

23 Distribution Functions (Def 1.2) Suppose our experiment has a nite number of outcomes. Let X be the random variable and Ω be the sample space. A distribution function for X is a real-valued function m whose domain is Ω and which satises: 1. m(ω) 0 for all outcomes ω Ω. 2. m(ω) = 1. ω Ω For any event E, the probability of E is P(E) = m(ω). ω E In particular, for the event with just one outcome {ω}, P({ω}) = m(ω).

24 Family Distribution Function Let's say in the past week, the phone rang six times: The dad picked it up three times. The mom picked it up twice. The kid picked it up once. m(dad) = 3/6 = 1/2. m(mom) = 2/6 = 1/3. m(kid) = 1/6. The probability that a parent picks up is: P({dad, mom}) = m(dad) + m(mom) = 3/6 + 2/6 = 5/6.

25 Family Distribution Function Let's say in the past week, the phone rang six times: The dad picked it up three times. The mom picked it up twice. The kid picked it up once. m(dad) = 3/6 = 1/2. m(mom) = 2/6 = 1/3. m(kid) = 1/6. The probability that a parent picks up is: P({dad, mom}) = m(dad) + m(mom) = 3/6 + 2/6 = 5/6.

26 Family Distribution Function Let's say in the past week, the phone rang six times: The dad picked it up three times. The mom picked it up twice. The kid picked it up once. m(dad) = 3/6 = 1/2. m(mom) = 2/6 = 1/3. m(kid) = 1/6. The probability that a parent picks up is: P({dad, mom}) = m(dad) + m(mom) = 3/6 + 2/6 = 5/6.

27 Uniform Distribution Functions (Def 1.3+) The uniform distribution function on a sample space Ω with n outcomes is the function m given by m(ω) = 1 n for each ω in Ω. Each outcome is equally likely. If E is an event, then P(E) = (number of outcomes in E) (1/n) number of outcomes in E =. n

28 Uniform Distribution Functions (Def 1.3+) The uniform distribution function on a sample space Ω with n outcomes is the function m given by m(ω) = 1 n for each ω in Ω. Each outcome is equally likely. If E is an event, then P(E) = (number of outcomes in E) (1/n) number of outcomes in E =. n

29 Flipping a Fair Coin If we ip a fair coin, we expect it to land Heads with probability 1/2 and Tails with probability 1/2. Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1

30 Flipping a Fair Coin If we ip a fair coin, we expect it to land Heads with probability 1/2 and Tails with probability 1/2. Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1

31 Flipping a Fair Coin If we ip a fair coin, we expect it to land Heads with probability 1/2 and Tails with probability 1/2. Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1

32 Flipping a Fair Coin If we ip a fair coin, we expect it to land Heads with probability 1/2 and Tails with probability 1/2. Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1

33 Flipping a Fair Coin If we ip a fair coin, we expect it to land Heads with probability 1/2 and Tails with probability 1/2. Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1

34 Let's use our denitions. Let H and T be the outcomes Heads and Tails. If the coin is fair, then the probability of getting H should equal the probability of getting T: P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1 2 m(h) = 1 m(h) = 1/2,

35 P({H}) = P({T }) m(h) = m(t ). Summing the distribution function over all outcomes should get 1: m(h) + m(t ) = 1 Substitute: m(h) + m(h) = 1 2 m(h) = 1 m(h) = 1/2, and m(t ) = m(h) = 1/2 as well. Hence P({H}) = P({T }) = 1/2.

36 @Home: Reading Note: page numbers refer to printed version. Add 8 to get page numbers in a PDF reader. An example of a nonuniform distribution function can be found on page 26 of the textbook. Notice that the sample space has outcomes that keep track of the number of heads. This is dierent from the sample space in Example 1.7 on page 19, where the sequence of heads and tails is recorded, which gave a uniform distribution function. Odds (pages 27-28) are another way to express probabilities. They are used in gambling. Example 1.13 on page 29 has a countably innite sample space. Take a look at the probability of getting a head after a nite number of tosses, and look at the probability that the coin comes up tails with every throw.

37 Homework Please visit Look for the syllabus for this class and read it. Please read Sections 1.1 and 1.2 (you can skip the historical remarks).

38 Help and Assistance I hold oce hours. Check the class syllabus for times and locations. If the times do not work, we can try to schedule an appointment. You can also me at nicholas.nguyen@uky.edu

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