Propability - Practice Materials for Statistics

Probability Random Experiments Sample Space A random experiment is an observational process whose results cannot be known in advance. For example, when a customer enters a Lexus dealership, will the customer buy a car or not? How much will the customer spend? The set of all possible outcomes (denoted S) is the sample space for the experiment. Some sample spaces can be enumerated easily, while others may be immense or impossible to enumerate. For example, when CitiBank makes a consumer loan, we might define a sample space with only two outcomes: S = {default, no default} The sample space describing a Walmart customer’s payment method might have four outcomes: S = {cash, debit card, credit card, check} The sample space to describe rolling a die has six outcomes: When two dice are rolled, the sample space consists of 36 outcomes, each of which is a pair: Event Probability What is ‘probability’? There are three distinct ways of assigning probability, listed in Table 5.1. Many people mix them up or use them interchangeably; however, each approach must be considered separately Empirical Approach Classical Approach Statisticians use the term a priori to refer to the process of assigning probabilities before we actually observe the event or try an experiment. When flipping a coin or rolling a pair of dice, we do not actually have to perform an experiment because the nature of the process allows us to envision the entire sample space. Instead, we can use deduction to determine P(A). This is the classical approach to probability. For example, in the sample space for the Sample Space for rolling two dice Subjective Approach A subjective probability reflects someone’s informed judgment about the likelihood of an event. The subjective approach to probability is needed when there is no repeatable random experiment. For example: • What is the probability that Ford’s new supplier of plastic fasteners will be able to meet the September 23 shipment deadline? • What is the probability that a new truck product program will show a return on investment of at least 10 percent? • What is the probability that the price of Ford’s stock will rise within the next 30 days? Exercises Rules of Probability Complement of an Event Complement of an Event A Union of two Events Intersection of two Events The intersection of two events A and B is the event consisting of all outcomes in the sample space S that are contained in both event A and event B. The intersection of A and B is denoted. General Law of addition The general law of addition says that the probability of the union of two events A and B is the sum of their probabilities less the probability of their intersection. Example: Mutually Exclusive Event Special Law of Addition Collective Exhaustive Events Conditional Probability General law of multiplication Example: Exercises: Independent Events When P(A) differs from P(A | B), the events are dependent. You can easily think of examples of dependence. For example, cell phone text messaging is more common among young people, while arteriosclerosis is more common among older people. Therefore, knowing a person’s age would affect the probability that the individual uses text messaging or has arteriosclerosis. Dependent events may be causally related, but statistical dependence does not prove cause and effect. It only means that knowing that event B has occurred will affect the probability that event A will occur. When knowing that event B has occurred does not affect the probability that event A will occur, then events A and B are independent. In other words, event A is independent of event B if the conditional probability P(A | B) is the same as the unconditional probability P(A); that is, if the probability of event A is the same whether event B occurs or not. For example, if text messaging among high school students is independent of gender, this means that knowing whether a student is a male or female does not change the probability that the student uses text messaging. If A and B are independent events, we can simplify the general law of multiplication (formula 5.10) to show that the joint probability of events A and B is the product of their individual probabilities. Recall the general law of multiplication: If A and B are independent, then we can substitute P(A) for P(A | B). The result is shown in formula 5.12, the special law of multiplication Contingency table Inspection of this table reveals that MBA graduates of the high-tuition schools do tend to have large salary gains (15 of the 67 schools) and that about half of the top-tier schools charge high tuition (32 of 67 schools). We can make more precise interpretations of these data by applying the concepts of probability. Margin Probabilities Tree Diagrams