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## Chapter 5

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**Chapter 5**Probability Created by Kathy Fritz**Can ultrasound accurately predict the gender of a baby?**The paper “The Use of Three-Dimensional Ultrasound for Fetal Gender Determination in the First Trimester” (The British Journal of Radiology [2003]: 448-451) describes a study of ultrasound gender prediction.An experienced radiologist looked at 159 first trimester ultrasound images and made a gender prediction for each one. When each baby was born, the ultrasound gender prediction was compared to the baby’s actual gender. This table summarizes the resulting data:**Notice that the gender prediction based on the ultrasound**image is NOT always correct. The paper also included gender predictions by a second radiologist, who looked at 154 first trimester ultrasound images.**Interpreting Probabilities**Probability Relative Frequency Law of Large Numbers Basic Properties**Probability**We often find ourselves in situations where the outcome is uncertain: When a ticketed passenger shows up at the airport, she faces two possible outcomes: (1) she is able to take the flight, or (2) she is denied a seat as a result of overbooking by the airline and must take a later flight. Based on her past experience, the passenger believes that the chance of being denied a seat is small or unlikely.**Subjective Approach to Probability**The subjective interpretation of probability is A probability of 1 A probability of 0 Because different people may have different subjective beliefs, they may assign different probabilitiesto the same outcome.**Relative Frequency Approach**In the relative frequency interpretationof probability, Relative frequency can be computed by:**A package delivery service promises 2-day delivery between 2**cities in California but is often able to deliver the packages in just 1 day. The company reports that the probability of next-day delivery is 0.3. Suppose that you track the delivery of packages shipped with this company. With each new package shipped, you could compute the relative frequency of packages shipped so far that have arrived in 1 day: Here is a graph displaying the relative frequencies for each of the first 15 packages shipped.**Here is a graph displaying the relative frequencies for**each of the first 50 packages shipped. Here is a graph displaying the relative frequencies for each of the first 1000 packages shipped.**Law of Large Numbers**Chance behavior is unpredictable in the short run, but has a regular and predictable patter in the long run.**ProbabilityModels**Descriptions of chance behavior contain two parts:**Example: Roll the Dice**Give a probability model for the chance process of rolling two fair, six-sided dice – one that’s red and one that’s blue.**Probability Models**Probability models allow us to find the probability of any collection of outcomes.**Example: Roll the Die**If A is any event, we write its probability as P(A). In the dice-rolling example, suppose we define event A as “sum is 5.” There are 4 outcomes that result in a sum of 5. Since each outcome has probability Suppose event B is defined as “sum is not 5.” What is P(B)?**All probability models must obey the following:**Some Basic Properties of Probability**A large auto center sells cars made by many different**manufacturers. Two of these are Honda and Toyota. Suppose: P(Honda) = 0.25 and P(Toyota) = 0.14 Consider the make of the next car sold. What is the probability that the next car sold is either a Honda or a Toyota?**4.**Some Basic Properties of Probability Recall the car dealership (P(Honda) = 0.25): What is the probability that the next car sold is not a Honda?**Computing Probabilities**Chance Experiment Sample Space Event Classical Approach to Probability**A chance experiment is**Chance Experiment Suppose two six-sided dice are rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection.**Sample Space**Consider a chance experiment to investigate whether men or women are more likely to choose a hybrid engine over a traditional internal combustion engine when purchasing a Honda Civic at a particular dealership. The type of vehicle purchased (hybrid or traditional) will be determined and the customer’s gender will be recorded. A list of all possible outcomes are:**Chance Experiment**Recall the situation in which a person purchases a Honda Civic: Sample space = {MH, FH, MT, FT} Identify the following events: traditional = female =**Classical Approach to Probability**When the outcomes in the sample space of a chance experiment are equally likely, the probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space:**Four students (Adam (A), Bettina (B), Carlos (C), and**Debra(D)) submitted correct solutions to a math contest that had two prizes. The contest rules specify that if more than two correct responses are submitted, the winners will be selected at random from those submitting correct responses. What is the sample space for selecting the two winners from the four correct responses? Because the winners are selected at random, the six possible outcomes are equally likely.**Four students (Adam (A), Bettina (B), Carlos (C), and**Debra(D)) submitted correct solutions to a math contest that had two prizes. The contest rules specify that if more than two correct responses are submitted, the winners will be selected at random from those submitting correct responses. Let E be the event that both selected winners are the same sex. What is the probability of E?**Four students (Adam (A), Bettina (B), Carlos (C), and**Debra(D)) submitted correct solutions to a math contest that had two prizes. The contest rules specify that if more than two correct responses are submitted, the winners will be selected at random from those submitting correct responses. Let F be the event that at least one of the selected winners is female. What is the probability of F?**Relative Frequency Approach to Probability**The probability of an event E, denoted by P(E), is defined to be the value approached by the relatively frequency of occurrence of E in a very long series of observations from a chance experiment. If the number of observations is large,**Suppose that you perform a chance experiment that consists**of flipping a cap from a 20-ounce bottle of soda and noting whether the cap lands with the top up or down. You carry out this chance experiment by flipping the cap 1000 times and record if it lands top up or top down. The cap lands top up 694 times.**Probabilities of More Complex Events**Union Intersection Complement Mutually Exclusive Events Independents Events**Consider the chance experiment that consists of selecting a**student at random from those enrolled at a particular college. There are 9000 students enrolled at the college Here are some possible events: F = event that the selected student is female O = event that the selected student is older than 30 A = event that the selected student favors the expansion of the athletic program S = event that the selected student is majoring is one of the lab sciences**Complement**The probability of EC can be computed from the probability of E as follows: Suppose that 4300 of the 9000 students favor the expansion of the athletic program. What is the probability of event A not occurring?**Intersection**Consider the events: O = event that the selected student is older than 30 S = event that the selected student is majoring is one of the lab science This table summaries the occurrence of these events:**Intersection**What is the probability of a randomly selected student is older than 30 AND is majoring in a lab science?**Union**Consider the events: O = event that the selected student is older than 30 A = event that the selected student favors the expansion of the athletic program This table summaries the occurrence of these events:**Union**What is the probability of a randomly selected student is older than 30 OR favors the expansion of the athletic program?**Two-Way Tables and Probability**Note, the previous example illustrates the fact that we can’t use the addition rule for mutually exclusive events unless the events have no outcomes in common. The Venn diagram below illustrates why. General Addition Rule for Two Events If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) – P(A and B)**Venn Diagrams and Probability**Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed.**Venn Diagrams and Probability**Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.**Venn Diagrams and Probability**Recall the example on gender and pierced ears. We can use a Venn diagram to display the information and determine probabilities. Define events A: is male and B: has pierced ears.**Hypothetical 1000**You can use tables to compute the probability of an intersection of two events and the probability of a union of two events. In many situations, you may ONLY know the probabilities of some events. In this case, it is often possible to create a “hypothetical 1000” table and then use the table to compute probabilities.**The report “TV Drama/Comedy Viewers and Health**Information” (www.cdc.gov/healthmarketing) describes a large survey that was conducted by the Centers for Disease Control (CDC). The CDC believed that the sample was representative of adult Americans. Let’s investigate these events (taken from questions on the survey): L = event that a randomly selected adult American reports learning something new about a health issue or disease from a TV show in the previous 6 months. F = event that a randomly selected adult American is female. Data from the survey were used to estimate the following probabilities:**CDC study continued**What is the probability that a randomly selected adult American has learned something new about a health issue or disease from a TV show in the previous 6 months or is female?**Let’s look at the hypothetical table once more.**Suppose: P (A) = 0.6, P (B C) = 0.7, and P (AB) = 0.2 What is the probability of A or B happening?**Mutually Exclusive Events**Sometimes people call the emergency 9-1-1 number to report situations that are not considered emergencies (such as to report a lost dog). Let two events be: M= event that the next call to 9-1-1 is for a medical emergency N= event that the next call to 9-1-1 is not considered an emergency Suppose that you know P(M) = 0.30 and P(N) = 0.20. Events M and N are mutually exclusive because the next call can’t be both a medical emergency and a call that is not considered an emergency.**P(M) = 0.30 and P(N) = 0.20**Mutually Exclusive Events**If E and F are mutually exclusive events, then**and Addition Rule for Mutually Exclusive Events**Independent Events**Suppose that you purchase a desktop computer system with a separate monitor and keyboard. Two possible events are: Event 1: The monitor needs service while under warranty. Event 2: The keyboard needs service while under warranty.**Dependent Events**Consider a university’s course registration process, which divides students into 12 priority groups. Overall, only 10% of all students receive all requested classes, but 75% of those in the first priority group receive all requested classes. You would say that the probability that a randomly selected student at this university receives all requested class is 0.10. However, if you know that the selected student is in the first priority group, you revise the probability that the student receives all requested classes to 0.75. These two events are said to be dependent events.**Multiplication Rule for Two Independent Events**More generally, if there are kindependent events, the probability that all the events occur is the product of all individual event probabilities.