Chapter 12: Introducing Probability PDF

CHAPTER 12: Introducing Probability In Chapter 12, we cover … The idea of probability The search for randomness Probability models Probability rules Finite probability models Continuous probability models Random variables Personal probability  The idea of probability Chance beh...

CHAPTER 12: Introducing Probability In Chapter 12, we cover … The idea of probability The search for randomness Probability models Probability rules Finite probability models Continuous probability models Random variables Personal probability  The idea of probability Chance behavior is unpredictable in the short run, but it has a regular and predictable pattern in the long run. RANDOMNESS AND PROBABILITY We call a phenomenon random if individual outcomes are uncertain but there is, nonetheless, a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.  Probability models Descriptions of chance behavior contain two parts: a list of possible outcomes and a probability for each outcome.  The sample space S of a random phenomenon is the set of all possible outcomes.  An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.  A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events.  Probability models Example: Give a probability model for the chance process of rolling two fair, six-sided dice―one that is red and one that is green. Sample space Since the dice are fair, each outcome is equally likely. 36 outcomes Each outcome has probability 1/36. Probability rules 1. Any probability is a number between 0 and 1. Any proportion is a number between 0 and 1, so any probability is also a number between 0 and 1. 2. All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly 1. 3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. 4. The probability that an event does not occur is 1 minus the probability that the event does occur. The probability that an event occurs and the probability that it does not occur always add to 1, or 100%. Probability rules The probability rules in formal language: Rule 1. The probability of any event satisfies Rule 2. If is the sample space in a probability model, Rule 3. Two events and are disjoint if they have no outcomes in common and, thus, can never occur together. If and are disjoint, This is the addition rule for disjoint events. Rule 4. For any event Probability rules (example) The first digits of numbers in legitimate financial records often follow a model known as Benford’s law. Call the first digit of a randomly chosen record X. Benford’s law gives the following probability model for X. First digit 1 2 3 4 5 6 7 8 9 (X) Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 (a) Show that this is a legitimate probability model. Each probability is between 0 and 1, and (b) Find the probability that the first digit for the chosen number is not a 9. Finite Probability Models One way to assign probabilities to events is to assign a probability to every individual outcome, then add these probabilities to find the probability of any event. This idea works well when there are only a finite (fixed and limited) number of outcomes. A probability model with a finite sample space is called a finite probability model. To assign probabilities in a finite model, list the probabilities of all the individual outcomes. These probabilities must be numbers between 0 and 1 that add to exactly 1. The probability of any event is the sum of the probabilities of the outcomes making up the event. Finite probability models are sometimes called discrete probability models. Statisticians often refer to finite probability models as discrete. The Benford’s law probability model was finite.  Continuous probability models Suppose we want to choose a number at random between 0 and 1, allowing any number between 0 and 1 as the outcome. We cannot assign probabilities to each individual value because there is an infinite continuum of possible values. A continuous probability model assigns probability as an area under a density curve. The area under the curve and above any range of values is the probability of an outcome in that range. Uniform Example: Find the probability Distribution of getting a random number that is less than or equal to 0.5 or greater than 0.8. Random variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what values can take and how to assign probabilities to those values. A finite random variable has a finite list of possible outcomes. Random variables that can take on any value in an interval, with probabilities given as areas under a density curve, are called continuous. Personal Probability A personal probability of an outcome is a number between 0 and 1 that expresses an individual’s judgment of how likely the outcome is. To be legitimate, a personal probability must obey Rules 1 to 4 of probability. It may not match another individual’s personal probability of an event.