Math 3339 Online – Week 2 Notes Section 2.1 – Sample Spaces and Events Definitions: An experiment is any activity or process whose outcome is subject to uncertainty. The sample space of an experiment, denoted by S (or simply S) is the set of all possible outcomes of that experiment. Ex: If we examine three fuses in sequence and note the result of each examination by either N when the fuse is not defective or D when the fuse is defective, then an outcome for the entire experiment is any sequence of N’s and D’s. Ex: Two gas stations are located at a certain intersection. Each one has six gas pumps. Consider the experiment in which the number of pumps in use at a particular time of day is determined for each of the stations. Identify the sample space. Def: An event is any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome. Ex: When the number of pumps in use at each of the two gas stations in the last example is observed, how many possible simple events are there? What would be a compound event? Set Theory: Definitions: 1. The complement of an event A, denoted by A’, is the set of all outcomes in S that are not contained in A. 2. The union of two events A and B, denoted by A ∪ B and read “A or B,” is the event consisting of all outcomes that are either in A or in B or in both. 3. The intersection of two events A and B, denoted by A ∩ B and read “A and B,” is the event consisting of all outcomes that are in both A and B. Ex: For the experiment in which the number of pumps in use at a single six-pump gas station is observed, let A = {0,1,2,3,4}, B = {3,4,5,6}, and C = {1,3,5} A' A∪ B A∪C A∩ B ( A ∩ C )' ( A ∪ B)' Def: Let ∅ denote the null event (the event consisting of no outcomes whatsoever). When A ∩ B = ∅ , A and B are said to be mutually exclusive or disjoint events. Section 2.2 - Axioms, Interpretations, and Properties of Probability Given a sample space, our goal is to assign to each event in the sample space a probability which will give us the chance that the event occurs. Axioms: 1. For any event A, P ( A) ≥ 0 . 2. P(S) = 1. 3. If A1 , A2 , A3 ,... is an infinite collection of disjoint events, then ∞ P ( A1 ∪ A2 ∪ A3 ∪ ...) = ∑ P ( Ai ) i =1 Proposition: P (∅ ) = 0 Ex: Consider testing batteries coming off an assembly line one by one until having a voltage within prescribed limits is found. What are the events in the sample space? (Use F for the event that a battery fails to meet the requirements and S for the event that a battery meets the requirements) More Properties of Probability: For any event A, P ( A) + P ( A ') = 1. So, For any event A, P ( A) ≤ 1 For any two events A and B, P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B ) Why is this true? In a certain residential suburb, 60% of all households get internet service from the local cable company, 80% get television service from that company, and 50% get both services from that company. If a household is randomly selected, what is the probability that it gets at least one of these services from the company? Ex: During off-peak hours a commuter train has five cars. Suppose a commuter is twice as likely to select the middle car (#3) as to select either adjacent car (#2 or #4), and is twice as likely to select either adjacent car as to select either end car (#1 or #5). Determine the probability that one of the three middle cars is selected. Equally Likely Outcomes: In an experiment with N equally likely outcomes, what is the probability of each outcome? Now for an event A, determine P(A). Section 2.3 - Counting Techniques When determining the probability of an event, we will need to be able to “count” the number outcomes in the event. Here are a series of counting rules to help us. Proposition: (The Product Rule) If the first element or object of an ordered pair can be selected pair can be selected n2 n1 ways, then the number of possible pairs is ways and the second element of the n1n2 . Ex: A homeowner doing some remodeling requires the services of both a plumber and an electrician. If there are 12 plumbers and 9 electricians available in the area, how many ways can the pair be chosen? Ex: A family has just moved to a new city and requires both an obstetrician and a pediatrician. There are two easily accessible medical clinic, each having two obstetricians and three pediatricians. If the family wishes to choose both from the same clinic, in how many ways can this be done? Generalized Product Rule: The number of ways of choosing a collection of k objects is n1n2 ...nk where ni represents the number of ways of choosing the i-th object. Ex: Suppose you are remodeling your kitchen and wish to purchase all new kitchen appliances. You need a stove, fridge, dishwasher, and microwave. Your local appliance store is having a sale on LG appliances if you purchase this brand exclusively. They offer 3 different LG stoves, 4 different LG fridges, 2 LG dishwashers, and 2 LG microwaves. How many different ways are there of purchasing one of each appliance? Permutations and Combinations: Definitions: An ordered subset (or list) is called a permutation. The number of permutations of size k that can be formed from the n individuals or objects in a group will be denoted by Pk ,n or in some texts n Pk . An unordered subset is called a combination. This is denoted Ck ,n or “n choose k”. ⎛n⎞ C or n k ⎜ ⎟ , and is read ⎝k ⎠ Proposition: Pk ,n = n! ( n − k )! ⎛n⎞ ⎝k ⎠ and ⎜ ⎟ = n! k !( n − k )! Ex: A particular iPod playlist contains 100 songs, 10 of which are by the Beatles. Suppose the shuffle feature is used to play the songs in random order. What is the probability that the first Beatles song heard is the fifth song played? Ex: A university warehouse has received a shipment of 25 printers, of which 10 are laser printers and 15 are inkjet models. If 6 of these 25 are selected at random to be checked by a particular technician, what is the probability that exactly 3 of those selected are laser printers? What is the probability that at least 3 laser printers are selected? What is the probability that at least one laser printer is selected? Section 2.4 - Conditional Probability Suppose a six-die is rolled. What is the probability of getting a 3? Suppose that we know that an odd number was rolled. What is the probability of getting a 3? The Conditional Probability of an event A given that event B has occurred is given by P ( A | B) = P ( A  B) P ( B) where P ( B ) ≠ 0 Ex: Suppose that a checkout line at the grocery store either has no waiting time (with probability 1/2), minor waiting time (with probability 1/3) , or considerable waiting time (with probability 1/6). If a customer approaches the checkout and someone is already there (meaning there will be waiting time) what is the probability that the waiting time will be considerable? Ex: Suppose that of all individuals buying a certain digital camera, 60% include an optional memory card in their purchase, 40% include an extra battery, and 30% include both a card and battery. Consider randomly selecting a buyer and let A = {memory card purchased} B = {battery purchased} Given that the selected individual purchased an extra battery, determine the probability that a memory card was also purchased. Ex: A news magazine publishes three columns entitled “Art” (A), “Books” (B), and “Cinema” (C). Reading habits of a randomly selected reader with respect to these columns are Determine the probability that the reader reads the art column if it is known that he read the books column. Determine the probability that he reads the art column given that he has read at least one of the other columns. Determine the probability the reader will read the art column given that he will read at least one column. The Law of Total Probability Theorem: Let A1 , A2 ,..., Ak be mutually exclusive and exhaustive events. Then for any event B, P ( B ) = P ( B | A1 ) P ( A1 ) + P ( B | A2 ) P ( A2 ) + ... + P ( B | Ak ) P ( Ak ) k = ∑ P ( B | Ai ) P ( Ai ) i =1 Ex: An individual has 3 different email accounts. Most of her messages, in fact 70%, come into account #1, whereas 20% come into account #2, and the remaining 10% come into account #3. Of the messages into account #1, only 1% are spam, whereas the corresponding percentages for accounts #2 and #3 are 2% and 5% respectively. What is the probability that a randomly selected message is spam? Bayes’ Theorem Bayes’ Theorem If a random experiment can result in k mutually exclusive and exhaustive outcomes A1 , A2 , A3 ,... Ak , then for an event B P ( Aj | B ) = P ( B | Aj ) P ( Aj ) P (B) = P ( B | Aj ) P ( Aj ) ∑ P(B | A ) P( A ) i i i Case: 2 A’s Case: 3 A’s Suppose that we know that 5 percent of the population have a certain disease. Suppose also, that no test for the disease is 100% accurate. A particular screening for the disease is proposed. In tests it is determined that this screening will be positive for a person without the disease 2% of the time, and the test will be negative for a person with the disease 8% of the time. Determine the probability that a person who has tested positive for the disease does not have the disease. Ex: Items in your inventory are produced at three different plants, 50% from plant 1, 30% from plant 2, and 20% from plant 3. In addition, each plant produces at different levels of quality. Plant 1 produces 5% defectives, plant 2 produces 7% defectives, and plant 3 produces 8% defectives. If an item from your inventory is found to be defective, what is the probability that is was produced in plant 1? Section 2.5 - Independence Independence: ( ) ( ) ( ) Two events are called independent if P A  B = P A P B . In this case, what are the conditional probabilities? Two events that are not independent are called dependent events. Ex: If we toss a nickel and dime at the same time, their outcomes are independent. Determine the probability that the nickel is heads and the dime is tails. Ex: Suppose that a machine has a vital component that fails with probability 0.01. In order to improve the probability that the machine runs properly, and second and third copy of this component is installed to run in parallel with the first. What is the probability that the machine runs properly? Ex: Consider a gas station with six pumps numbered 1,2,…,6, and let Ei denote the simple event that a randomly selected customer uses pump i (i = 1,2,…,6). Suppose that P ( E1 ) = P ( E6 ) = 0.10 P ( E2 ) = P ( E5 ) = 0.15 Define events A, B, and C by A = {2,4,6}, B = {1,2,3}, C = {2,3,4,5}. Which events are independent? dependent? P ( E3 ) = P ( E4 ) = 0.25