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Transcript of Ch5 Probability
3We now take a break from the statistical process. Why? In
Chapter 1, we mentioned that inferential statistics uses methods
that generalize results obtained from a sample to the population
and measures their reliability. But how can we measure their
reliability? It turns out that the methods we use to generalize
results from a sample to a population are based on probability
and probability models. Probability is a measure of the likelihood
that something occurs. This part of the course will focus on meth-
ods for determining probabilities.
CHAPTER 6Discrete ProbabilityDistributions
CHAPTER 7The NormalProbabilityDistribution
Probability and ProbabilityDistributions
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Outline5.1 Probability Rules
5.2 The Addition Rule andComplements
5.3 Independence and theMultiplication Rule
5.4 Conditional Probabilityand the GeneralMultiplication Rule
5.5 Counting Techniques
5.6 Putting It Together: WhichMethod Do I Use?
5.7 Bayess Rule (on CD)
Have you ever watched a sportingevent on television in which the an-
nouncer cites an obscure statistic?Where do these numbers
come from? Well, pretendthat you are the statisti-cian for your favorite
sports team. Your jobis to compile strangeor obscure probabili-
ties regarding your favoriteteam and a competingteam. See the Decisions
project on page 326.
PUTTING IT TOGETHERIn Chapter 1, we learned the methods of collecting data. In Chapters 2 through 4, we learned how to summa-rize raw data using tables, graphs, and numbers. As far as the statistical process goes, we have discussed thecollecting, organizing, and summarizing parts of the process.
Before we can proceed with the analysis of data, we introduce probability, which forms the basis of in-ferential statistics. Why? Well, we can think of the probability of an outcome as the likelihood of observingthat outcome. If something has a high likelihood of happening, it has a high probability (close to 1). If some-thing has a small chance of happening, it has a low probability (close to 0). For example, in rolling a singledie, it is unlikely that we would roll five straight sixes, so this result has a low probability. In fact, the proba-bility of rolling five straight sixes is 0.0001286. So, if we were playing a game that entailed throwing a singledie, and one of the players threw five sixes in a row, we would consider the player to be lucky (or a cheater)because it is such an unusual occurrence. Statisticians use probability in the same way. If something occursthat has a low probability, we investigate to find out whats up.
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Probability is a measure of the likelihood of a random phenomenon or chancebehavior. Probability describes the long-term proportion with which a certainoutcome will occur in situations with short-term uncertainty.
The long-term predictability of chance behavior is best understood through asimple experiment. Flip a coin 100 times and compute the proportion of heads ob-served after each toss of the coin. Suppose the first flip is tails, so the proportion of
heads is the second flip is heads, so the proportion of heads is the third flip is
heads, so the proportion of heads is and so on. Plot the proportion of heads versus
the number of flips and obtain the graph in Figure 1(a). We repeat this experiment with the results shown in Figure 1(b).
Section 5.1 Probability Rules 259
Note to InstructorIf you like, you can print out and distrib-ute the Preparing for This Section quizlocated in the Instructors ResourceCenter. The purpose of the quiz is to verify that the students have the prereq-uisite knowledge for the section.
Note to InstructorProbabilities can be expressed as frac-tions, decimals, or percents. You maywant to give a brief review of how to con-vert from one form to the other.
5.1 PROBABILITY RULESPreparing for This Section Before getting started, review the following: Relative frequency (Section 2.1, p. 68)
In Other WordsProbability describes how likely it is thatsome event will happen. If we look at theproportion of times an event has occurredover a long period of time (or over a largenumber of trials), we can be more certainof the likelihood of its occurrence.
0 50 100
Number of Flips
1st flip tails
2nd flip heads
3rd flip heads
Objectives 1 Apply the rules of probabilities2 Compute and interpret probabilities using the empirical method3 Compute and interpret probabilities using the classical method4 Use simulation to obtain data based on probabilities5 Recognize and interpret subjective probabilities
0 50 100
Number of Flips
The Law of Large NumbersAs the number of repetitions of a probability experiment increases, the propor-tion with which a certain outcome is observed gets closer to the probability ofthe outcome.
Looking at the graphs in Figures 1(a) and (b), we notice that in the short term(fewer flips of the coin) the observed proportion of heads is different and unpredictablefor each experiment.As the number of flips of the coin increases,however,both graphstend toward a proportion of 0.5. This is the basic premise of probability. Probabilitydeals with experiments that yield random short-term results or outcomes yet reveallong-term predictability. The long-term proportion with which a certain outcome isobserved is the probability of that outcome. So we say that the probability of observing
a head is or 50% or 0.5 because, as we flip the coin more times, the proportion of
heads tends toward This phenomenon is referred to as the Law of Large Numbers.12
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The Law of Large Numbers is illustrated in Figure 1. For a few flips of the coin,the proportion of heads fluctuates wildly around 0.5, but as the number of flips in-creases, the proportion of heads settles down near 0.5. Jakob Bernoulli (a major con-tributor to the field of probability) believed that the Law of Large Numbers wascommon sense. This is evident in the following quote from his text Ars Conjectandi:For even the most stupid of men, by some instinct of nature, by himself and withoutany instruction, is convinced that the more observations have been made, the lessdanger there is of wandering from ones goal.
In probability, an experiment is any process with uncertain results that can berepeated.The result of any single trial of the experiment is not known ahead of time.However, the results of the experiment over many trials produce regular patternsthat enable us to predict with remarkable accuracy. For example, an insurance com-pany cannot know ahead of time whether a particular 16-year-old driver will be in-volved in an accident over the course of a year. However, based on historicalrecords, the company can be fairly certain that about three out of every ten 16-year-old male drivers will be involved in a traffic accident during the course of a year.Therefore, of the 816,000 male 16-year-old drivers (816,000 repetitions of the exper-iment), the insurance company is fairly confident that about 30%, or 244,800, of thedrivers will be involved in an accident. This prediction forms the basis for establish-ing insurance rates for any particular 16-year-old male driver.
We now introduce some terminology that we will need to study probability.
260 Chapter 5 Probability
Note to InstructorIn-class activity: Have each student flip acoin three times. Use the results to findthe probability of a head. Repeat thisexperiment a few times. Use the cumula-tive results to illustrate the Law of LargeNumbers. You may also want to use theprobability applet to demonstrate theLaw of Large Numbers. See Problem 57.
Definitions The sample space, S, of a probability experiment is the collection of all possibleoutcomes.
An event is any collection of outcomes from a probability experiment.An eventmay consist of one outcome or more than one outcome. We will denote eventswith one outcome, sometimes called simple events, In general, events aredenoted using capital letters such as E.
The following example illustrates these definitions.
EXAMPLE 1 Identifying Events and the Sample Space of a Probability Experiment
Problem: A probability experiment consists of rolling a single fair die.(a) Identify the outcomes of the probability experiment.(b) Determine the sample space.(c) Define the event
Approach: The outcomes are the possible results of the experiment. The samplespace is a list of all possible outcomes.
Solution(a) The outcomes from rolling a single fair die are a
a a aa and
(b) The set of all possible outcomes forms the sample space,There are 6 outcomes in the sample space.
(c) The event E = roll an even number = 52, 4, 66.S = 51, 2, 3, 4, 5, 66.
e6 = rolling a six = 566.five = 556,e5 = rollingfour = 546,e4 = rollingthree = 536,e3 = rollingtwo = 526,e2 = rolling
one = 516,e1 = rolling
E = roll an even number.
In Other WordsAn outcome is the result of one trial of a probability experiment. The samplespace is a list of all possible results ofa probability experiment.
A fair die is on