Math HL Chapter 12 Probability Name: Read the notes and fill in any blanks. Work through the ALL of the examples. Self-Check your own progress by rating where you are. # Learning Targets Lesson I have NO Idea! Which describes you best? I know I know some of most of this! this! I ve got this! 5.2 5.3 I can use the concepts of trial, outcome, equally likely outcomes, sample space (U) and event. I can find the probability of an event A. I can find and use the complementary events as A and A (not A) I can use Venn diagrams, tree diagrams and tables of outcomes to solve problems. I can find the probability of combined events and mutually exclusive events. B: A: B: A: B: A: B: A: B: A: 5.4 I can find conditional probability I can find the probability of independent events. I can use Bayes theorem for a maximum of three events. Graphs and Visual Representations: B: A: B: A: B: A: 1
Math HL Chapter 12 Probability In drawing conclusions we use inferential statistics, which use probability as one of its tools. Random variables: Probability distributions: 12.1: Randomness Probability is the study of Events that are random are 12.2: Basic Definitions: We use the term experiment to describe data obtained by observing either uncontrolled events in nature or controlled situations in a laboratory. An experiment is A random experiment is an experiment A description of a random phenomenon in the language of mathematics is called a probability model. When we toss a coin, we cannon know the outcome in advance. But, what do we know? 2
The sample space S of a random experiment (or phenomenon) is the set of all possible outcomes. Ex 1: Toss a coin twice (or two coins once) and record the results. Write is the sample space? Ex 2: Toss a coin twice (or two coins once) and count the number of heads showing. What is the sample space? A simple event is A event is an Ex 3: When rolling a standard six-sided die, what are the sets of event A observe an odd number, and event B observe a number less than 5. Use a Venn diagram to help visualize an experiment. The rectangle represents the entire sample space Each circle represents each event Ex 4: space? Suppose we choose one card at random from a deck of 52 playing cards, what is the sample 3
Ex 5: Toss a coin three times and record the results. Show the event observing two heads as a Venn diagram. Tree Diagrams, Tables and Grids: Tree Diagram in an experiment to check the blood types of patients, the experiment can be represented in a tree diagram. Same info in a table: In a 2-D grid 4
Ex 6: Two tetrahedral dice, one blue and one yellow, are rolled. List the elements of the following events: T = {3 appears on at least one die} B = {the blue die is a 3} S = {sum of the dice is a six} Project: Silence the Violence (We will do this at the beginning of the school year.) 5
12.3: Probability Assignments: Equally likely outcomes: Frequency theory: In all theories, probability is on a scale of 0% to 100%. Probability and chance are synonymous. Probability Rules: Rule 1: probability is a number The probability of any event A satisfies If the probability of any event is If the probability of any event is, then, then Rule 2: all possible outcomes must have a Rule 3: If two events have the probability that one or the other occurs is Two events that have and can are called or This is the Rule 4: The event that contains the is called the and is denoted by Ex 7: Data for traffic violations was collected in a certain country and a summary is given below. What is the probability that the offender is A) In the youngest age group? B) Between 21 & 40? C) Younger than 40? 6
Ex 8: It is a striking fact that when people create codes for their cellphones, the first digits follow distributions very similar to the following one: A) Find the probabilities of the following three events: A = {first digit is 1} B = {first digit is more than 5} C = {first digit is an odd number} B) Find the probability that the first digit is 1) 1 or greater than 5 2) Not 1 3) An odd number or a number larger than 5 Equally Likely Outcomes: Suppose in ex 8, all of the digits are considered to be equally likely to happen, then the table would be: Find the probabilities of the following three events: A = {first digit is 1} B = {first digit is more than 5} C = {first digit is an odd number} 2-D grids are also very helpful tools used to visualize 2-stage or sequential probability models. For example, rolling a normal unbiased die twice. 7
Geometric Probability: You want to randomly shoot a dart at a circular target. What is the probability of hitting the central part? Ex 9: Lydia and Rania agreed to meet at the museum quarter between 12:00 noon and 1:00 p.m. The first person to arrive will wait 15 minutes. If the second person does not show up, the first person will leave and they meet afterwards. Assuming that their arrivals are at random, what is the probability that they meet? Probability Calculation for Equally Likely Outcomes Using Counting Principles: Ex 10: In a group of 18 students, eight are females. What is the probability of choosing five students A) With all girls? B) With three girls and two boys? C) With at least one boy? 8
Ex 11: A deck of playing cards has 52 cards. In a game, the player is given five cards. Find the probability of the player having: A) Three cards of one denomination and two cards of another (three 7s and two Js for example) A player was given the following hand: cards. Find the probability of the player having: She decided to change the last two a. Three cards of one denomination and two cards of another. b. Four queens 9
12.4: Operations with Events: In example #8, we looked at the events: B = {first digit is more than 5} C = {first digit is an odd number} The two events were also noted at not disjoint. If events are not disjoint, then there must be something in common. Thus B I C = Since the intersection contains elements, the sets B and C are not mutually exclusive. Find P(B I C) State B U C = Notice that 7 & 9 are in both sets, thus potentially may be counted twice. To find the probability of B U C we must make sure not to count them twice. What can be done? Rule 5: For any two events 10
Rule 6: The simple multiplication rule. If two events then Consider the following situation: At Upper Arlington High School, 55% of the students are male. It is also known that the percentage of musicians among males and females in this school is the same, 22%. What is the probability of selecting a student at random from this population and the student is a male musician? Two events A and B are independent if The multiplication rule for independent events: if two events A and B are Then 11
Ex 12: Reconsider the situation with the traffic light at the beginning of this chapter. The probability that I find the light green is 30%. What is the probability that I find it green on two consecutive days? Ex 13: Computers bought from a HP require repairs quite frequently. It is estimated that 17% of computers bought from the company require one repair job during the first month of purchase, 7% will need repairs twice during the first month, and 4% require three or more repairs. A) What is the probability that a computer chosen at random from HP will need: a) No repairs? b) No more than one repair? c) Some repair? B) If you buy two such computers, what is the probability that: a. Neither will require repair? b. Both will need repair? Conditional Probability: In probability, conditioning means incorporating new restrictions on the outcome of an experiment (updating probabilities to take into account new information). Ex 14: A public health department wanted to study the smoking behavior of high school students. They interviewed 768 students from grades 10-12 and asked them about their smoking habits. They categorized the students into three categories: smokers (more than 1 pack per week); occasional smokers (less than 1 pack per week), and non-smokers. If we select a student at random, what is the probability that we select: A) A girl? B) A male smoker? C) A non-smoker? 12
This is an example of the Multiplication Rule of any two events A and B. Multiplication Rule: Given any events A and B, the probability that both events happen is given by Ex 15: In a psychology lab, researchers are studying the color preferences of young children. Six green toys and four red toys (identical apart from color) are placed in a container. The child is asked to select two toys at random. What is the probability that the child chooses two red toys? Use a tree diagram When the conditional probability of A given B is Why does this formula make sense? 13
Ex 16: In an experiment to study the phenomenon of color blindness, researchers collected information concerning 1000 people in a small town and categorized them according to color blindness and gender. Here is a summary of the findings. Ex 17: AUA, a national airline, is known for its punctuality. The probability that a regularly scheduled flight departs on time is P(D) = 0.83, the probability that it arrives on time is P(A) = 0.92, and the probability that it arrives and departs on time, P A I D ( ) = 0.78. Find the probability that a flight A) Arrives on time given that it departed on time. B) Departs on time given that it arrived on time. Independence: Two events are independent if Two events are independent if and only if either or Otherwise, the events are Ex 18: Take another look at the AUA situation in example 17. Are the events of arriving on time (A) and departing on time (D) independent? 14
Ex 19: In many countries, the police stop drivers on suspicion of drunk driving. The stopped drivers are given a breath test, a blood test, a blood test or both. In a country where this problem is vigorously dealt with, the police records show the following: 81% of the drivers stopped are given a breath test, 40% a blood test, and 25% both tests A) What is the probability that a suspected driver is given i) A test? ii) Exactly one test? iii) No test? B) Are giving the two tests, independent? Ex 20: Jane and Kate frequently play tennis with each other. When Jane serves first, she wins 60% of the time, and the same pattern occurs with Kate. They alternate the serve, of course. They usually play for a prize, which is a chocolate bar. The first one who loses on her serve will have to buy the chocolate. Jane serves first: A) Find the probability that Jane pays on her second serve. B) Find the probability that Jane eventually pays for the chocolate. C) Find the probability that Kate pays for the chocolate. 15
Ex 21: A target for a dart game is shown. The radius of the board is 40 cm and it is divided into three regions as shown. You score 2 points if you hit the center, 1 point for the middle region and 0 points for the outer region. A) What is the probability of scoring a 1 in one attempt? B) What is the probability of scoring a 2 in one attempt? C) How many attempts are necessary so that the probability of scoring at least one 2 is at least 50%? 16
12.5: Bayes Theorem: Tests with high precision rates are open to error. Bayes theorem helps us understand and analyze the results of such tests. Suppose we have 9 indistinguishable boxes with blue and red balls. There are three types: Type A contains 2 balls each one red and one blue; Type B contains 3 balls each one blue and two red; and Type C contains 4 balls each one blue and 3 red. We mix up the boxes and choose one at random and then pick a ball from that box. What is the probability that you can guess the box type that it was drawn from if the ball is red? 17
Ex 22: 60% of the students at a university are male and 40% are female. Records show that 30% of the males have IB diplomas while 75% of the females have IB diplomas. A student is selected and found to have a diploma. What is the probability that the student is a female? Bayes Theorem Simple Case: Let S be a sample space with E 1 and E 2 mutually exclusive events that partition this sample space. Let F be a non-empty event in this sample space. Ex 23: Lie detectors (polygraphs) are not considered reliable in court. They are, however, administered to employees in sensitive positions. One such test gives a positive reading (the person is lying) when a person is lying 88% of the time and a negative reading (person telling the truth) when a person is telling the truth 86% of the time. In a security-related question, the vast majority of subjects have no reason to lie so that 99% of the subjects will tell the truth. An employee produces a positive response on the polygraph. What is the probability that this employee is actually telling the truth? 18
Bayes Theorem General Case: Let S be a sample space with E 1, E 2,..., E n mutually exclusive events that partition this sample space. Let F be a non-empty event in this sample space. Then: Ex 24: A paper factory produces high-quality paper using two machines. Like any process, the final product is checked for quality. 98% of the time machine A produces paper that confirms to the accepted norms, 97.5% of machine B s paper conforms to norms. Machine A produces 60% of the paper in this company. You pick a paper at random and it does not conform to the norms what is the change it was produced by A? Ex 25: In many countries the law requires that a driver s license is withdrawn if he/she is found to have more than 0.05% blood alcohol concentration (BAC). Suppose that the people use a test that will correctly identify a drunk driver (testing positive) 99% of the time, and will correctly identify a sober driver (testing negative) 99% of the time. Let s assume that 0.5% of the drivers in your city drive under the influence of alcohol. We want to know the probability that given a positive test a driver is actually drunk. 19
Ex 26: A computer manufacturer receives hard disks from three different suppliers. Marco supplies 40% of the disks; Berto supplies 25% while Lukas supplies the rest. Disks from Marco have a defective rate of 4%, those from Berto 3%, while Lukas have a 5% rate. A) A disk is checked at random. What is the probability that it is defective? B) A disk is checked and found defective. What is the probability that it was supplied by Lukas? 20