Introducing Probability - virtual.yosemite.cc.ca.usvirtual.yosemite.cc.ca.us/jcurl/Math 134 3 s/chap09np.pdf · We might suspect that a coin has probability 0.5 of coming up heads

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Introducing Probability In this chapter we cover. . .The idea of probability

Thinking about randomness

Probability models

Probability rules

Assigning probabilities:finite number of outcomes

Assigning probabilities:intervals of outcomes

Normal probability models

Random variables

Personal probability

Why is probability, the science of chance behavior, needed to understand statis-tics, the science of data? Let’s look at a typical sample survey.

EXAMPLE 9.1 Do you lotto?

What proportion of all adults bought a lottery ticket in the past 12 months? Wedon’t know, but we do have the results of a Gallup Poll. Gallup took a sample of1523 adults. It happens that 868 of the people in the sample bought tickets. Theproportion who bought tickets was

sample proportion = 8681523

= 0.57 (that is, 57%)

Because all adults had the same chance to be among the chosen 1523, it seems rea-sonable to use this 57% as an estimate of the unknown proportion in the population.It’s a fact that 57% of the sample bought lottery tickets—we know because Gallupasked them. We don’t know what percent of all adults bought tickets, but we estimatethat about 57% did. This is a basic move in statistics: use a result from a sample toestimate something about a population.

What if Gallup took a second random sample of 1523 adults? The new sam-ple would have different people in it. It is almost certain that there would notbe exactly 868 positive responses. That is, Gallup’s estimate of the proportionof adults who bought a lottery ticket will vary from sample to sample. Couldit happen that one random sample finds that 57% of adults recently bought alottery ticket and a second random sample finds that only 37% had done so?Random samples eliminate bias from the act of choosing a sample, but they canstill be wrong because of the variability that results when we choose at random.

223



224 CHAPTER 9 � Introducing Probability

If the variation when we take repeat samples from the same population is toogreat, we can’t trust the results of any one sample.

This is where we need facts about probability to make progress in statistics.Because Gallup uses impersonal chance to choose its samples, the laws of prob-ability govern the behavior of the samples. Gallup says that when the GallupPoll takes a sample, the probability is 0.95 that the estimate from the sam-ple comes within ±3 percentage points of the truth about the population ofall adults. The first step toward understanding this statement is to understandwhat “probability 0.95” means. Our purpose in this chapter is to understand thelanguage of probability, but without going into the mathematics of probabilitytheory.

The idea of probabilityTo understand why we can trust random samples and randomized compara-tive experiments, we must look closely at chance behavior. The big fact thatemerges is this: chance behavior is unpredictable in the short run but has aregular and predictable pattern in the long run.

Toss a coin, or choose an SRS. The result can’t be predicted in advance,because the result will vary when you toss the coin or choose the sample re-peatedly. But there is still a regular pattern in the results, a pattern that emergesclearly only after many repetitions. This remarkable fact is the basis for the ideaof probability.

(Super Stock)

EXAMPLE 9.2 Coin tossing

When you toss a coin, there are only two possible outcomes, heads or tails. Figure 9.1shows the results of tossing a coin 5000 times twice. For each number of tosses from1 to 5000, we have plotted the proportion of those tosses that gave a head. Trial A(solid line) begins tail, head, tail, tail. You can see that the proportion of heads forTrial A starts at 0 on the first toss, rises to 0.5 when the second toss gives a head,then falls to 0.33 and 0.25 as we get two more tails. Trial B, on the other hand, startswith five straight heads, so the proportion of heads is 1 until the sixth toss.

The proportion of tosses that produce heads is quite variable at first. Trial A startslow and Trial B starts high. As we make more and more tosses, however, the propor-tion of heads for both trials gets close to 0.5 and stays there. If we made yet a thirdtrial at tossing the coin a great many times, the proportion of heads would againsettle down to 0.5 in the long run. We say that 0.5 is the probability of a head. Theprobability 0.5 appears as a horizontal line on the graph.

APPLET

The Probability applet animates Figure 9.1. It allows you to choose the prob-ability of a head and simulate any number of tosses of a coin with that probabil-ity. Experience shows that the proportion of heads gradually settles down closeto the probability. Equally important, it also shows that the proportion in asmall or moderate number of tosses can be far from the probability. Probabilitydescribes only what happens in the long run.



225The idea of probability

1.0

0.9

0.8

0.7

0.6

0.2

0.3

0.4

0.5

0.1

0.0

1 105 50 100 500 1000 5000

Number of tosses

Pro

po

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f h

ead

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Figure 9.1 The proportion of tosses of a coin that give a head changes as we makemore tosses. Eventually, however, the proportion approaches 0.5, the probability of ahead. This figure shows the results of two trials of 5000 tosses each.

“Random” in statistics is not a synonym for “haphazard” but a descriptionof a kind of order that emerges only in the long run. We often encounter theunpredictable side of randomness in our everyday experience, but we rarely seeenough repetitions of the same random phenomenon to observe the long-termregularity that probability describes. You can see that regularity emerging inFigure 9.1. In the very long run, the proportion of tosses that give a head is 0.5.This is the intuitive idea of probability. Probability 0.5 means “occurs half thetime in a very large number of trials.”

We might suspect that a coin has probability 0.5 of coming up heads just be-cause the coin has two sides. As Exercises 9.3 and 9.4 illustrate, such suspicionsare not always correct. The idea of probability is empirical. That is, it is basedon observation rather than theorizing. Probability describes what happens invery many trials, and we must actually observe many trials to pin down a prob-ability. In the case of tossing a coin, some diligent people have in fact madethousands of tosses.

EXAMPLE 9.3 Some coin tossers

The French naturalist Count Buffon (1707–1788) tossed a coin 4040 times. Result:2048 heads, or proportion 2048/4040 = 0.5069 for heads.

Around 1900, the English statistician Karl Pearson heroically tossed a coin24,000 times. Result: 12,012 heads, a proportion of 0.5005.




While imprisoned by the Germans during World War II, the South African math-ematician John Kerrich tossed a coin 10,000 times. Result: 5067 heads, a proportionof 0.5067.

RANDOMNESS AND PROBABILITY

We call a phenomenon random if individual outcomes are uncertain butthere is nonetheless a regular distribution of outcomes in a large numberof repetitions.The probability of any outcome of a random phenomenon is theproportion of times the outcome would occur in a very long series ofrepetitions.

APPLY YOUR KNOWLEDGE

9.1 Deal a straight. A “straight” in poker is a hand of five cards whoseranks are consecutive when you arrange them in order, such as 8, 9, 10,Jack, Queen. You read in a book on poker that the probability of beingdealt a straight is 4/1000. Explain carefully what this means.

9.2 Three of a kind. “Three of a kind” in poker is a hand of five cards thatcontains three of the same rank, such as three 7’s, along with two othercards that are not 7’s and that differ from each other. You read in a bookon poker that the probability of being dealt three of a kind is 1/50.Explain why this does not mean that if you deal 50 hands, exactly onewill contain three of a kind.

9.3 Nickels spinning. Hold a nickel upright on its edge under yourforefinger on a hard surface, then snap it with your other forefinger sothat it spins for some time before falling. Based on 50 spins, estimate theprobability of heads.

9.4 Tossing a thumbtack. Toss a thumbtack on a hard surface 100 times.How many times did it land with the point up? What is the approximateprobability of landing point up?

Does God play dice?

Few things in the world aretruly random in the sense thatno amount of information willallow us to predict theoutcome. We could inprinciple apply the laws ofphysics to a tossed coin, forexample, and calculatewhether it will land heads ortails. But randomness does ruleevents inside individual atoms.Albert Einstein didn’t like thisfeature of the new quantumtheory. “I shall never believethat God plays dice with theworld,” said the great scientist.Eighty years later, it appearsthat Einstein was wrong.

Thinking about randomnessThat some things are random is an observed fact about the world. The outcomeof a coin toss, the time between emissions of particles by a radioactive source,and the sexes of the next litter of lab rats are all random. So is the outcome of arandom sample or a randomized experiment. Probability theory is the branch ofmathematics that describes random behavior. Of course, we can never observea probability exactly. We could always continue tossing the coin, for example.Mathematical probability is an idealization based on imagining what wouldhappen in an indefinitely long series of trials.



227Thinking about randomness

The best way to understand randomness is to observe random behavior—notonly the long-run regularity but the unpredictable results of short runs. You cando this with physical devices, as in Exercises 9.3 and 9.4, but computer simula-tions (imitations) of random behavior allow faster exploration. Exercises 9.5and 9.6 suggest some simulations of random behavior using the Probabilityapplet. There are more applet simulations in the Media Exercises at the endof this chapter. As you explore randomness, remember:� You must have a long series of independent trials. That is, the outcome of independence

one trial must not influence the outcome of any other. Imagine a crookedgambling house where the operator of a roulette wheel can stop it whereshe chooses—she can prevent the proportion of “red” from settling downto a fixed number. These trials are not independent.� The idea of probability is empirical. Computer simulations start with

given probabilities and imitate random behavior, but we can estimate areal-world probability only by actually observing many trials.� Nonetheless, computer simulations are very useful because we need long

runs of trials. In situations such as coin tossing, the proportion of anoutcome often requires several hundred trials to settle down to theprobability of that outcome. The kinds of physical random devicessuggested in the exercises are too slow for this. Short runs give only roughestimates of a probability.


9.5 Random digits. The table of random digits (Table B) was produced by arandom mechanism that gives each digit probability 0.1 of being a 0.

APPLET

APPLET(a) What proportion of the first 50 digits in the table are 0s? This

proportion is an estimate, based on 50 repetitions, of the trueprobability, which in this case is known to be 0.1.

(b) The Probability applet can imitate random digits. Set the probabilityof heads in the applet to 0.1. Check “Show true probability” to showthis value on the graph. A head stands for a 0 in the random digittable and a tail stands for any other digit. Simulate 200 digits (40 ata time—don’t click “Reset”). If you kept going forever, presumablyyou would get 10% heads. What was the result of your 200 tosses?

9.6 How many tosses to get a head? When we toss a coin, experienceshows that the probability (long-term proportion) of a head is close to1/2. Suppose now that we toss the coin repeatedly until we get a head.What is the probability that the first head comes up in an odd number oftosses (1, 3, 5, and so on)?

APPLET

(a) Start with an actual coin. Repeat this experiment 10 times, andkeep a record of the number of tosses needed to get a head on eachof your 10 trials. Use your results to estimate the probability that thefirst head appears on an odd-numbered toss.




(b) Now use the Probability applet. Set the probability of heads to 0.5.Toss coins one at a time until the first head appears. Do this 50 times(click “Reset” after each trial). What is your estimate of theprobability that the first head appears on an odd toss?

9.7 Probability says . . . . Probability is a measure of how likely an event isto occur. Match one of the probabilities that follow with each statementof likelihood given. (The probability is usually a more exact measure oflikelihood than is the verbal statement.)

0 0.01 0.3 0.6 0.99 1

(a) This event is impossible. It can never occur.(b) This event is certain. It will occur on every trial.(c) This event is very unlikely, but it will occur once in a while in a long

sequence of trials.(d) This event will occur more often than not.

Probability modelsGamblers have known for centuries that the fall of coins, cards, and dice dis-plays clear patterns in the long run. The idea of probability rests on the ob-served fact that the average result of many thousands of chance outcomes canbe known with near certainty. How can we give a mathematical description oflong-run regularity?

To see how to proceed, think first about a very simple random phenomenon,tossing a coin once. When we toss a coin, we cannot know the outcome inadvance. What do we know? We are willing to say that the outcome will beeither heads or tails. We believe that each of these outcomes has probability1/2. This description of coin tossing has two parts:� A list of possible outcomes.� A probability for each outcome.Such a description is the basis for all probability models. Here is the basic vo-cabulary we use.

PROBABILITY MODELS

The sample space S of a random phenomenon is the set of all possibleoutcomes.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 randomphenomenon consisting of two parts: a sample space S and a way ofassigning probabilities to events.

The sample space S can be very simple or very complex. When we toss acoin once, there are only two outcomes, heads and tails. The sample space is



229Probability models

Figure 9.2 The 36 possible outcomes in rolling two dice. If the dice are carefullymade, all of these outcomes have the same probability.

S = {H, T}. When Gallup draws a random sample of 1523 adults, the samplespace contains all possible choices of 1523 of the 210 million adults in thecountry. This S is extremely large. Each member of S is a possible sample, whichexplains the term sample space.

(Super Stock)

EXAMPLE 9.4 Rolling dice

Rolling two dice is a common way to lose money in casinos. There are 36 possibleoutcomes when we roll two dice and record the up-faces in order (first die, seconddie). Figure 9.2 displays these outcomes. They make up the sample space S. “Rolla 5” is an event, call it A, that contains four of these 36 outcomes:

A = { }

Gamblers care only about the number of pips on the up-faces of the dice. The samplespace for rolling two dice and counting the pips is

S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}Comparing this S with Figure 9.2 reminds us that we can change S by changing thedetailed description of the random phenomenon we are describing.


9.8 Sample space. Choose a student at random from a large statistics class.Describe a sample space S for each of the following. (In some cases youmay have some freedom in specifying S.)(a) Ask how much time the student spent studying during the past

24 hours.(b) Ask how much money in coins (not bills) the student is carrying.(c) Record the student’s letter grade at the end of the course.(d) Ask whether the student did or did not take a math class in each of

the two previous years of school.




9.9 Sample space. In each of the following situations, describe a samplespace S for the random phenomenon.(a) A basketball player shoots four free throws. You record the sequence

of hits and misses.(b) A basketball player shoots four free throws. You record the number

of baskets she makes.9.10 Sample space. In each of the following situations, describe a sample

space S for the random phenomenon. In some cases you have somefreedom in specifying S, especially in setting the largest and the smallestvalue in S.(a) A study of healthy adults measures the weights of adult women.(b) The Physicians’ Health Study asked 11,000 physicians to take an

aspirin every other day and observed how many of them had a heartattack in a five-year period.

(c) In a test of a new package design, you drop a carton of a dozen eggsfrom a height of 1 foot and count the number of broken eggs.

Probability rulesThe true probability of any outcome—say, “roll a 5 when we toss two dice”—can be found only by actually tossing two dice many times, and then only ap-proximately. How then can we describe probability mathematically? Ratherthan try to give “correct” probabilities, we start by laying down facts that mustbe true for any assignment of probabilities. These facts follow from the ideaof probability as “the long-run proportion of repetitions on which an eventoccurs.”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 between0 and 1. An event with probability 0 never occurs, and an event withprobability 1 occurs on every trial. An event with probability 0.5 occursin half the trials in the long run.

2. All possible outcomes together must have probability 1. Because someoutcome must occur on every trial, the sum of the probabilities for allpossible outcomes must be exactly 1.

3. The probability that an event does not occur is 1 minus the probabilitythat the event does occur. If an event occurs in (say) 70% of all trials,it fails to occur in the other 30%. The probability that an event occursand the probability that it does not occur always add to 100%, or 1.

4. If two events have no outcomes in common, the probability that oneor the other occurs is the sum of their individual probabilities. If oneevent occurs in 40% of all trials, a different event occurs in 25% of alltrials, and the two can never occur together, then one or the other occurson 65% of all trials because 40% + 25% = 65%.

We can use mathematical notation to state Facts 1 to 4 more concisely.Capital letters near the beginning of the alphabet denote events. If A is any



231Probability rules

event, we write its probability as P(A). Here are our probability facts in formallanguage. As you apply these rules, remember that they are just another formof intuitively true facts about long-run proportions.

PROBABILITY RULES

Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.Rule 2. If S is the sample space in a probability model, then P(S) = 1.Rule 3. For any event A,

P (A does not occur) = 1 − P(A)

Rule 4. Two events A and B are disjoint if they have no outcomes incommon and so can never occur simultaneously. If A and B aredisjoint,

P(A or B) = P(A) + P(B)

This is the addition rule for disjoint events.

The probability ofrain is . . .

You work all week. Then itrains on the weekend. Canthere really be a statisticaltruth behind our perceptionthat the weather is against us?At least on the east coast ofthe United States, the answeris “Yes.” Going back to 1946, itseems that Sundays receive22% more precipitation thanMondays. The likelyexplanation is that thepollution from all thoseworkday cars and trucks formsthe seeds for raindrops—withjust enough delay to cause rainon the weekend.

EXAMPLE 9.5 Benford’s law

Faked numbers in tax returns, payment records, invoices, expense account claims,and many other settings often display patterns that aren’t present in legitimaterecords. Some patterns, like too many round numbers, are obvious and easily avoidedby a clever crook. Others are more subtle. It is a striking fact that the first digits ofnumbers in legitimate records often follow a distribution known as Benford’s law.Here it is:1

First digit 1 2 3 4 5 6 7 8 9

Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

The probabilities assigned to first digits by Benford’s law are all between 0 and 1. Benford’s lawThey add to 1 because the first digits listed make up the sample space S. (Note thata first digit can’t be 0.)

The probability that a first digit is anything other than 1 is, by Rule 3,

P (not a 1) = 1 − P (first digit is 1)= 1 − 0.301 = 0.699

That is, if 30.1% of first digits are 1s, the other 69.9% are not 1s.“First digit is a 1” and “first digit is a 2” are disjoint events. So the addition rule

(Rule 4) says

P (1 or 2) = P (1) + P (2)= 0.301 + 0.176 = 0.477

About 48% of first digits in data governed by Benford’s law are 1s or 2s. Fraudulentrecords generally have many fewer 1s and 2s.




EXAMPLE 9.6 Probabilities for rolling dice

Figure 9.2 (page 229) displays the 36 possible outcomes of rolling two dice. Whatprobabilities should we assign to these outcomes?

Casino dice are carefully made. Their spots are not hollowed out, which wouldgive the faces different weights, but are filled with white plastic of the same den-sity as the colored plastic of the body. For casino dice it is reasonable to assign thesame probability to each of the 36 outcomes in Figure 9.2. Because all 36 outcomestogether must have probability 1 (Rule 2), each outcome must have probability 1/36.

Gamblers are often interested in the sum of the pips on the up-faces. What is theprobability of rolling a 5? Because the event “roll a 5” contains the four outcomesdisplayed in Example 9.4, the addition rule (Rule 4) says that its probability is

P (roll a 5) = P ( ) + P ( ) + P ( ) + P ( )

= 136

+ 136

+ 136

+ 136

= 436

= 0.111

What about the probability of rolling a 7? In Figure 9.2 you will find six outcomesfor which the sum of the pips is 7. The probability is 6/36, or about 0.167.


9.11 Rolling a soft 4. A “soft 4” in rolling two dice is a roll of 1 on one dieand 3 on the other. If you roll two dice, what is the probability of rollinga soft 4? Of rolling a 4?

9.12 Preparing for the GMAT. A company that offers courses to preparewould-be MBA students for the Graduate Management Admission Test(GMAT) has the following information about its customers: 20% arecurrently undergraduate students in business; 15% are undergraduatestudents in other fields of study; 60% are college graduates who arecurrently employed; and 5% are college graduates who are not employed.(a) Is this a legitimate assignment of probabilities to customer

backgrounds? Why?(b) What percent of customers are currently undergraduates?

9.13 Forests in Missouri. What happens to trees over a five-year period? Astudy lasting more than 30 years found these probabilities for a randomlychosen 12-inch-diameter tree in the Ozark Highlands of Missouri: stayin the 12-inch class, 0.686; move to the 14-inch class, 0.256.2 Theremaining trees die during the five-year period. What is the probabilitythat a tree dies?

Assigning probabilities: finite number of outcomesExamples 9.5 and 9.6 illustrate one way to assign probabilities to events: assigna probability to every individual outcome, then add these probabilities to find



233Assigning probabilities: finite number of outcomes

the probability of any event. If such an assignment is to satisfy the rules of prob-ability, the probabilities of all the individual outcomes must sum to exactly 1.

PROBABILITIES IN A FINITE SAMPLE SPACE

Assign a probability to each individual outcome. These probabilitiesmust be numbers between 0 and 1 and must have sum 1.The probability of any event is the sum of the probabilities of theoutcomes making up the event.

Really random digits

For purists, the RANDCorporation long agopublished a book titled OneMillion Random Digits. Thebook lists 1,000,000 digits thatwere produced by a veryelaborate physicalrandomization and really arerandom. An employee ofRAND once told me that thisis not the most boring bookthat RAND has everpublished.

EXAMPLE 9.7 Random digits versus Benford’s law

You might think that first digits in financial records are distributed “at random”among the digits 1 to 9. The 9 possible outcomes would then be equally likely. Thesample space for a single digit is

S = {1, 2, 3, 4, 5, 6, 7, 8, 9}Call a randomly chosen first digit X for short. The probability model for X is com-pletely described by this table:

First digit X 1 2 3 4 5 6 7 8 9

Probability 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

The probability that a first digit is equal to or greater than 6 is

P (X ≥ 6) = P (X = 6) + P (X = 7) + P (X = 8) + P (X = 9)

= 19

+ 19

+ 19

+ 19

= 49

= 0.444

Note that this is not the same as the probability that a random digit is strictly greaterthan 6, P (X > 6). The outcome X = 6 is included in the event {X ≥ 6} and isomitted from {X > 6}.

Compare this with the probability of the same event among first digits from fi-nancial records that obey Benford’s law. A first digit V chosen from such records hasthis distribution:

First digit V 1 2 3 4 5 6 7 8 9

Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

The probability that this digit is equal to or greater than 6 is

P (V ≥ 6) = 0.067 + 0.058 + 0.051 + 0.046 = 0.222

Benford’s law allows easy detection of phony financial records that have been basedon randomly generated numbers. Such records tend to have too few first digits of 1and 2 and too many first digits of 6 or greater.




Outcome Model 1

1/7

1/7

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Model 2

1/3

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1/3

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1

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2

Figure 9.3 Four assignmentsof probabilities to the sixfaces of a die, for Exercise9.14.


9.14 Rolling a die. Figure 9.3 displays several assignments of probabilities tothe six faces of a die. We can learn which assignment is actually accuratefor a particular die only by rolling the die many times. However, some ofthe assignments are not legitimate assignments of probability. That is,they do not obey the rules. Which are legitimate and which are not? Inthe case of the illegitimate models, explain what is wrong.

9.15 Benford’s law. If first digits follow Benford’s law (Examples 9.5 and9.7), consider these events:

A = {first digit is 7 or greater}B = {first digit is odd}

(a) What outcomes make up event A? What is P(A)?(b) What outcomes make up event B? What is P(B)?(c) What outcomes make up the event “A or B”? What is P(A or B)?

Why is this probability not equal to P(A) + P(B)?9.16 Race and ethnicity. The 2000 census allowed each person to choose

from a long list of races. That is, in the eyes of the Census Bureau,you belong to whatever race you say you belong to. “Hispanic/Latino”is a separate category; Hispanics may be of any race. If we choose aresident of the United States at random, the 2000 census gives theseprobabilities:

Hispanic Not Hispanic

Asian 0.000 0.036Black 0.003 0.121White 0.060 0.691Other 0.062 0.027



235Assigning probabilities: intervals of outcomes

(a) Verify that this is a legitimate assignment of probabilities.(b) What is the probability that a randomly chosen American is

Hispanic?(c) Non-Hispanic whites are the historical majority in the United

States. What is the probability that a randomly chosen American isnot a member of this group?

Assigning probabilities: intervals of outcomesA software random number generator is designed to produce a number between0 and 1 chosen at random. It is only a slight idealization to consider any numberin this range as a possible outcome. The sample space is then

S = {all numbers between 0 and 1}Call the outcome of the random number generator Y for short. How can we as-sign probabilities to such events as {0.3 ≤ Y ≤ 0.7}? As in the case of selectinga random digit, we would like all possible outcomes to be equally likely. But wecannot assign probabilities to each individual value of Y and then add them,because there are infinitely many possible values.

We use a new way of assigning probabilities directly to events—as areas undera density curve. Any density curve has area exactly 1 underneath it, correspond-ing to total probability 1. We first met density curves as models for data inChapter 3 (page 58).

EXAMPLE 9.8 Random numbers

The random number generator will spread its output uniformly across the entireinterval from 0 to 1 as we allow it to generate a long sequence of numbers. The resultsof many trials are represented by the uniform density curve shown in Figure 9.4. Thisdensity curve has height 1 over the interval from 0 to 1. The area under the curve is1, and the probability of any event is the area under the curve and above the eventin question.

Height = 1

Area = 0.4 Area = 0.5 Area = 0.2

0 0.3 0.7 1 0 0.5 0.8 1

(a) P(0.3 ≤ Y ≤ 0.7) (b) P(Y ≤ 0.5 or Y > 0.8)

Figure 9.4 Probability as area under a density curve. This uniform density curvespreads probability evenly between 0 and 1.




As Figure 9.4(a) illustrates, the probability that the random number generatorproduces a number between 0.3 and 0.7 is

P (0.3 ≤ Y ≤ 0.7) = 0.4

because the area under the density curve and above the interval from 0.3 to 0.7 is 0.4.The height of the curve is 1 and the area of a rectangle is the product of height andlength, so the probability of any interval of outcomes is just the length of the interval.Similarly,

P (Y ≤ 0.5) = 0.5P (Y > 0.8) = 0.2

P (Y ≤ 0.5 or Y > 0.8) = 0.7

Notice that the last event consists of two nonoverlapping intervals, so the total areaabove the event is found by adding two areas, as illustrated by Figure 9.4(b). Thisassignment of probabilities obeys all of our rules for probability.


9.17 Random numbers. Let X be a random number between 0 and 1produced by the idealized random number generator described inExample 9.8 and Figure 9.4. Find the following probabilities:(a) P (0 ≤ X ≤ 0.4)(b) P (0.4 ≤ X ≤ 1)(c) P (0.3 ≤ X ≤ 0.5)

9.18 The probability that X = 0.5. Let X be a random number between 0and 1 produced by the idealized random number generator described inExample 9.8 and Figure 9.4. Find the following probabilities as areasunder the density function:(a) P (X < 0.5)(b) P (X ≤ 0.5)(c) Compare your two answers: what must be the probability that X is

exactly equal to 0.5? This is always true when we assign probabilitiesas areas under a density curve.

9.19 Adding random numbers. Generate two random numbers between 0and 1 and take Y to be their sum. The sum Y can take any value between0 and 2. The density curve of Y is the triangle shown in Figure 9.5.

0 1 2

Height = 1Figure 9.5 The densitycurve for the sum of tworandom numbers, for Exercise9.19. This density curvespreads probability between 0and 2.



237Random variables

(a) Verify by geometry that the area under this curve is 1.(b) What is the probability that Y is less than 1? (Sketch the density

curve, shade the area that represents the probability, then find thatarea. Do this for (c) also.)

(c) What is the probability that Y is less than 0.5?

Normal probability modelsAny density curve can be used to assign probabilities. The density curves thatare most familiar to us are the Normal curves. So Normal distributions areprobability models. There is a close connection between a Normal distributionas an idealized description for data and a Normal probability model. If we lookat the heights of all young women, we find that they closely follow the Normaldistribution with mean µ = 64 inches and standard deviation σ = 2.7 inches.That is a distribution for a large set of data. Now choose one young woman atrandom. Call her height X. If we repeat the random choice very many times,the distribution of values of X is the same Normal distribution.

EXAMPLE 9.9 The height of young women

What is the probability that a randomly chosen young woman has height between68 and 70 inches?

The height X of the woman we choose has the N(64, 2.7) distribution. Findthe probability by standardizing and using Table A, the table of standard Normalprobabilities. We will reserve capital Z for a standard Normal variable.

P (68 ≤ X ≤ 70) = P(

68 − 642.7

≤ X − 642.7

≤ 70 − 642.7

)

= P (1.48 ≤ Z ≤ 2.22)= 0.9868 − 0.9306 = 0.0562

Figure 9.6 shows the areas under the standard Normal curve. The calculation is thesame as those we did in Chapter 3. Only the language of probability is new.

Random variablesExamples 9.7 to 9.9 use a shorthand notation that is often convenient. InExample 9.9, we let X stand for the result of choosing a woman at random andmeasuring her height. We know that X would take a different value if we madeanother random choice. Because its value changes from one random choice toanother, we call the height X a random variable.

RANDOM VARIABLE

A random variable is a variable whose value is a numerical outcome of arandom phenomenon.The probability distribution of a random variable X tells us what valuesX can take and how to assign probabilities to those values.




1.48 2.22

Standard Normal curve Probability = 0.0562

Figure 9.6 The probability in Example 9.9 as an area under the standard Normalcurve.

We usually denote random variables by capital letters near the end of thealphabet, such as X or Y . Of course, the random variables of greatest inter-est to us are outcomes such as the mean x of a random sample, for which wewill keep the familiar notation. There are two main types of random variables,corresponding to two ways of assigning probability: discrete and continuous.

EXAMPLE 9.10 Discrete and continuous random variables

The random variable X in Example 9.7 (page 233) has as its possible values the wholenumbers {1, 2, 3, 4, 5, 6, 7, 8, 9}. The distribution of X assigns a probability to eachof these outcomes. Random variables that have a finite list of possible outcomes arecalled discrete.discrete random variable

Compare the random variable Y in Example 9.8 (page 235). The values of Y fillthe entire interval of numbers between 0 and 1. The probability distribution of Y isgiven by its density curve, shown in Figure 9.4. Random variables that can take onany value in an interval, with probabilities given as areas under a density curve, arecontinuous randomcalled continuous.variable


9.20 Grades in an accounting course. North Carolina State Universityposts the grade distributions for its courses online.3 Students inAccounting 210 in the spring 2001 semester received 18% A’s, 32% B’s,34% C’s, 9% D’s, and 7% F’s. Choose an Accounting 210 student atrandom. To “choose at random” means to give every student the samechance to be chosen. The student’s grade on a four-point scale (withA = 4) is a discrete random variable X.



239Personal probability

The value of X changes when we repeatedly choose students atrandom, but it is always one of 0, 1, 2, 3, or 4. Here is the distributionof X:

Value of X 0 1 2 3 4

Probability 0.07 0.09 0.34 0.32 0.18

(a) Say in words what the meaning of P (X ≥ 3) is. What is thisprobability?

(b) Write the event “the student got a grade poorer than C” in terms ofvalues of the random variable X. What is the probability of thisevent?

9.21 Rolling the dice. Example 9.6 describes the assignment of probabilitiesto the 36 possible outcomes of tossing two dice. Take the randomvariable X to be the sum of the values on the two up-faces. Example 9.6shows that P (X = 5) = 4/36 and that P (X = 7) = 6/36. Give thecomplete probability distribution of the discrete random variable X.That is, list the possible values of X and say what the probability of eachvalue is. Start from the 36 outcomes displayed in Figure 9.2.

9.22 Iowa Test scores. The Normal distribution with mean µ = 6.8 andstandard deviation σ = 1.6 is a good description of the Iowa Testvocabulary scores of seventh-grade students in Gary, Indiana. If we usethis model, the score Y of a randomly chosen student is a continuousrandom variable. Figure 3.1 (page 57) pictures the density function.(a) Write the event “the student chosen has a score of 10 or higher” in

terms of Y .(b) Find the probability of this event.

What are the odds?

Gamblers often express chancein terms of odds rather thanprobability. Odds of A to Bagainst an outcome means thatthe probability of thatoutcome is B/(A+ B). So“odds of 5 to 1” is another wayof saying “probability 1/6.” Aprobability is always between 0and 1, but odds range from 0 toinfinity. Although odds aremainly used in gambling, theygive us a way to make verysmall probabilities clearer.“Odds of 999 to 1” may beeasier to understand than“probability 0.001.”

Personal probability*We began our discussion of probability with one idea: the probability of an out-come of a random phenomenon is the proportion of times that outcome wouldoccur in a very long series of repetitions. This idea ties probability to actual out-comes. It allows us, for example, to estimate probabilities by simulating randomphenomena. Yet we often meet another, quite different, idea of probability.

EXAMPLE 9.11 Joe and the Chicago Cubs

Joe sits staring into his beer as his favorite baseball team, the Chicago Cubs, losesanother game. The Cubbies have some good young players, so let’s ask Joe, “What’sthe chance that the Cubs will go to the World Series next year?” Joe brightens up.“Oh, about 10%,” he says.

Does Joe assign probability 0.10 to the Cubs’ appearing in the World Series?The outcome of next year’s pennant race is certainly unpredictable, but we can’t

*This section is optional.




reasonably ask what would happen in many repetitions. Next year’s baseball seasonwill happen only once and will differ from all other seasons in players, weather, andmany other ways. If probability measures “what would happen if we did this manytimes,” Joe’s 0.10 is not a probability. Probability is based on data about many repeti-tions of the same random phenomenon. Joe is giving us something else, his personaljudgment.

Although Joe’s 0.10 isn’t a probability in our usual sense, it gives usefulinformation about Joe’s opinion. More seriously, a company asking, “Howlikely is it that building this plant will pay off within five years?” can’t employan idea of probability based on many repetitions of the same thing. Theopinions of company officers and advisors are nonetheless useful information,and these opinions can be expressed in the language of probability. These arepersonal probabilities.

PERSONAL PROBABILITY

A personal probability of an outcome is a number between 0 and 1 thatexpresses an individual’s judgment of how likely the outcome is.

Just as Rachel’s opinion about the Cubs may differ from Joe’s, the opinions ofseveral company officers about the new plant may differ. Personal probabilitiesare indeed personal: they vary from person to person. Moreover, a personalprobability can’t be called right or wrong. If we say, “In the long run, this coinwill come up heads 60% of the time,” we can find out if we are right by actuallytossing the coin several thousand times. If Joe says, “I think the Cubs have a10% chance of going to the World Series next year,” that’s just Joe’s opinion.Why think of personal probabilities as probabilities? Because any set of personalprobabilities that makes sense obeys the same basic Rules 1 to 4 that describeany legitimate assignment of probabilities to events. If Joe thinks there’s a 10%chance that the Cubs will go to the World Series, he must also think that there’sa 90% chance that they won’t go. There is just one set of rules of probability,even though we now have two interpretations of what probability means.


9.23 Will you have an accident? The probability that a randomly chosendriver will be involved in an accident in the next year is about 0.2. Thisis based on the proportion of millions of drivers who have accidents.“Accident” includes things like crumpling a fender in your owndriveway, not just highway accidents.(a) What do you think is your own probability of being in an accident

in the next year? This is a personal probability.(b) Give some reasons why your personal probability might be a more

accurate prediction of your “true chance” of having an accidentthan the probability for a random driver.



241Chapter 9 Exercises

(c) Almost everyone says their personal probability is lower than therandom driver probability. Why do you think this is true?

Chapter 9 SUMMARY

A random phenomenon has outcomes that we cannot predict but thatnonetheless have a regular distribution in very many repetitions.The probability of an event is the proportion of times the event occurs inmany repeated trials of a random phenomenon.A probability model for a random phenomenon consists of a sample space Sand an assignment of probabilities P.

The sample space S is the set of all possible outcomes of the randomphenomenon. Sets of outcomes are called events. P assigns a number P(A) toan event A as its probability.Any assignment of probability must obey the rules that state the basicproperties of probability:1. 0 ≤ P(A) ≤ 1 for any event A.

2. P (S) = 1.3. For any event A, P(A does not occur) = 1 − P(A).4. Addition rule: Events A and B are disjoint if they have no outcomes in

common. If A and B are disjoint, then P(A or B) = P(A) + P(B).When a sample space S contains finitely many possible values, a probabilitymodel assigns each of these values a probability between 0 and 1 such that thesum of all the probabilities is exactly 1. The probability of any event is thesum of the probabilities of all the values that make up the event.A sample space can contain all values in some interval of numbers.A probability model assigns probabilities as areas under a density curve. Theprobability of any event is the area under the curve above the values thatmake up the event.A random variable is a variable taking numerical values determined by theoutcome of a random phenomenon. The probability distribution of a randomvariable X tells us what the possible values of X are and how probabilities areassigned to those values.A random variable X and its distribution can be discrete or continuous.A discrete random variable has finitely many possible values. Its distributiongives the probability of each value. A continuous random variable takes allvalues in some interval of numbers. A density curve describes the probabilitydistribution of a continuous random variable.

Chapter 9 EXERCISES

9.24 Nickels falling over. You may feel that it is obvious that the probabilityof a head in tossing a coin is about 1/2 because the coin has two faces.




Such opinions are not always correct. Exercise 9.3 asked you to spin anickel rather than toss it—that changes the probability of a head. Nowtry another variation. Stand a nickel on edge on a hard, flat surface.Pound the surface with your hand so that the nickel falls over. What isthe probability that it falls with heads upward? Make at least 50 trials toestimate the probability of a head.

9.25 Got money? Choose a student at random and record the number ofdollars in bills (ignore change) that he or she is carrying. Give areasonable sample space S for this random phenomenon. (We don’tknow the largest amount that a student could reasonably carry, so youwill have to make a choice in stating the sample space.)

(Galen Rowell/CORBIS)

9.26 Land in Canada. Statistics Canada says that the land area of Canada is9,094,000 square kilometers. Of this land, 4,176,000 square kilometersare forested. Choose a square kilometer of land in Canada at random.(a) What is the probability that the area you choose is forested?(b) What is the probability that it is not forested?

9.27 Deaths on the job. Government data on job-related deaths assign asingle occupation for each such death that occurs in the United States.The data show that the probability is 0.134 that a randomly chosendeath was agriculture-related, and 0.119 that it was manufacturing-related. What is the probability that a death was either agriculture-related or manufacturing-related? What is the probability that the deathwas related to some other occupation?

9.28 Foreign language study. Choose a student in grades 9 to 12 at randomand ask if he or she is studying a language other than English. Here isthe distribution of results:

Language Spanish French German All others None

Probability 0.26 0.09 0.03 0.03 0.59

(a) Explain why this is a legitimate probability model.(b) What is the probability that a randomly chosen student is studying a

language other than English?(c) What is the probability that a randomly chosen student is studying

French, German, or Spanish?9.29 Car colors. Choose a new car or light truck at random and note its

color. Here are the probabilities of the most popular colors for vehiclesmade in North America in 2001:4

Color Silver White Black Blue Red Green

Probability 0.210 0.156 0.112 0.112 0.099 0.076




(a) What is the probability that the vehicle you choose has any colorother than the six listed?

(b) What is the probability that a randomly chosen vehicle is eithersilver or white?

9.30 Colors of M&M’s. If you draw an M&M candy at random from a bagof the candies, the candy you draw will have one of seven colors. Theprobability of drawing each color depends on the proportion of eachcolor among all candies made. Here is the distribution for milkchocolate M&M’s:5

Color Purple Yellow Red Orange Brown Green Blue

Probability 0.2 0.2 0.2 0.1 0.1 0.1 ?

(a) What must be the probability of drawing a blue candy?(b) What is the probability that you do not draw a brown candy?(c) What is the probability that the candy you draw is either yellow,

orange, or red?9.31 More M&M’s. Caramel M&M candies are equally likely to be brown,

red, yellow, green, orange, or blue. What is the probability for eachcolor?

9.32 Legitimate probabilities? In each of the following situations, statewhether or not the given assignment of probabilities to individualoutcomes is legitimate, that is, satisfies the rules of probability. If not,give specific reasons for your answer.(a) When a coin is spun, P (H) = 0.55 and P (T) = 0.45.(b) When two coins are tossed, P (HH) = 0.4, P (HT) = 0.4,

P (TH) = 0.4, and P (TT) = 0.4.(c) When a die is rolled, the number of spots on the up-face has

P (1) = 1/2, P (4) = 1/6, P (5) = 1/6, and P (6) = 1/6.

(Neil Beer/Photo Disc/PictureQuest)

9.33 Who goes to Paris? Abby, Deborah, Mei-Ling, Sam, and Roberto workin a firm’s public relations office. Their employer must choose two ofthem to attend a conference in Paris. To avoid unfairness, the choicewill be made by drawing two names from a hat. (This is an SRS ofsize 2.)(a) Write down all possible choices of two of the five names. This is the

sample space.(b) The random drawing makes all choices equally likely. What is the

probability of each choice?(c) What is the probability that Mei-Ling is chosen?(d) What is the probability that neither of the two men (Sam and

Roberto) is chosen?




9.34 Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36.The slots 0 and 00 are colored green, 18 of the others are red, and 18 areblack. The dealer spins the wheel and at the same time rolls a small ballalong the wheel in the opposite direction. The wheel is carefullybalanced so that the ball is equally likely to land in any slot when thewheel slows. Gamblers can bet on various combinations of numbers andcolors.(a) What is the probability of any one of the 38 possible outcomes?

Explain your answer.(b) If you bet on “red,” you win if the ball lands in a red slot. What is

the probability of winning?(c) The slot numbers are laid out on a board on which gamblers place

their bets. One column of numbers on the board contains allmultiples of 3, that is, 3, 6, 9, . . . , 36. You place a “column bet” thatwins if any of these numbers comes up. What is your probability ofwinning?

9.35 Birth order. A couple plans to have three children. There are 8possible arrangements of girls and boys. For example, GGB means thefirst two children are girls and the third child is a boy. All 8arrangements are (approximately) equally likely.(a) Write down all 8 arrangements of the sexes of three children. What

is the probability of any one of these arrangements?(b) Let X be the number of girls the couple has. What is the probability

that X = 2?(c) Starting from your work in (a), find the distribution of X. That is,

what values can X take, and what are the probabilities for eachvalue?

9.36 Living in San Jose. Let the random variable X be the number of roomsin a randomly chosen owner-occupied housing unit in San Jose,California. Here is the distribution of X:6

Rooms X 1 2 3 4 5 6 7 8 9 10

Probability 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035

(a) Is the random variable X discrete or continuous? Why?(b) Express “the unit has no more than 2 rooms” in terms of X. What is

the probability of this event?(c) Express the event {X > 5} in words. What is its probability?

9.37 Living in San Jose, continued. The previous exercise gives thedistribution of the number of rooms in owner-occupied housing in SanJose, California. Here is the distribution for rented housing:




Rooms Y 1 2 3 4 5 6 7 8 9 10

Probability 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003

What are the most important differences between the distributions ofthe random variables X and Y (owner-occupied and rented housing)?Compare at least two probabilities for X and Y to justify your answer.

9.38 How many cars in the garage? Choose an American household atrandom and let the random variable X be the number of cars (includingSUVs and light trucks) they own. Here is the probability model if weignore the few households that own more than 5 cars:

Number of cars X 0 1 2 3 4 5

Probability 0.09 0.36 0.35 0.13 0.05 0.02

(a) Verify that this is a legitimate discrete distribution.(b) Say in words what the event {X ≥ 1} is. Find P (X ≥ 1).(c) A housing company builds houses with two-car garages. What

percent of households have more cars than the garage can hold?9.39 Unusual dice. Nonstandard dice can produce interesting distributions

of outcomes. You have two balanced, six-sided dice. One is a standarddie, with faces having 1, 2, 3, 4, 5, and 6 spots. The other die has threefaces with 0 spots and three faces with 6 spots. Find the probabilitydistribution for the total number of spots Y on the up-faces when youroll these two dice. (Hint: Start with a picture like Figure 9.2 for thepossible up-faces. Label the three 0 faces on the second die 0a, 0b, 0c inyour picture, and similarly distinguish the three 6 faces.)

9.40 Random numbers. Many random number generators allow users tospecify the range of the random numbers to be produced. Suppose thatyou specify that the random number Y can take any value between 0and 2. Then the density curve of the outcomes has constant heightbetween 0 and 2, and height 0 elsewhere.(a) Is the random variable Y discrete or continuous? Why?(b) What is the height of the density curve between 0 and 2? Draw a

graph of the density curve.(c) Use your graph from (b) and the fact that probability is area under

the curve to find P (Y ≤ 1).9.41 Polling women. Suppose that 47% of all adult women think they do

not get enough time for themselves. An opinion poll interviews 1025randomly chosen women and records the sample proportion who don’tfeel they get enough time for themselves. Call this proportion V . It willvary from sample to sample if the poll is repeated. The distribution of




the random variable V is approximately Normal with mean 0.47 andstandard deviation about 0.016. Sketch this Normal curve and use it toanswer the following questions.(a) The truth about the population is 0.47. In what range will the

middle 95% of all sample results fall?(b) What is the probability that the poll gets a sample in which fewer

than 45% say they do not get enough time for themselves?

9.42 More random numbers. Find these probabilities as areas under thedensity curve you sketched in Exercise 9.40.(a) P (0.5 < Y < 1.3).(b) P (Y ≥ 0.8).

9.43 NAEP math scores. Scores on the National Assessment of EducationalProgress 12th-grade mathematics test for the year 2000 wereapproximately Normal with mean 300 points (out of 500 possible) andstandard deviation 35 points. Let Y stand for the score of a randomlychosen student. Express each of the following events in terms of Y anduse the 68–95–99.7 rule to give the approximate probability.(a) The student has a score above 300.(b) The student’s score is above 370.

Figure 9.7 A tetrahedron.Exercises 9.44 and 9.46 askabout probabilities for dicehaving this shape.

9.44 Tetrahedral dice. Psychologists sometimes use tetrahedral dice to studyour intuition about chance behavior. A tetrahedron (Figure 9.7) is apyramid with 4 faces, each a triangle with all sides equal in length. Labelthe 4 faces of a tetrahedral die with 1, 2, 3, and 4 spots. Give aprobability model for rolling such a die and recording the number ofspots on the down-face. Explain why you think your model is at leastclose to correct.

9.45 Playing Pick 4. The Pick 4 games in many state lotteries announce afour-digit winning number each day. The winning number is essentiallya four-digit group from a table of random digits. You win if your choicematches the winning digits. Suppose your chosen number is 5974.(a) What is the probability that your number matches the winning

number exactly?(b) What is the probability that your number matches the digits in the

winning number in any order?9.46 More tetrahedral dice. Tetrahedral dice are described in Exercise 9.44.

Give a probability model for rolling two such dice. That is, write downall possible outcomes and give a probability to each. (Figure 9.2 mayhelp you.) What is the probability that the sum of the down-faces is 5?

9.47 Playing Pick 4, continued. The Wisconsin version of Pick 4 pays out$5000 on a $1 bet if your number matches the winning number exactly.It pays $200 on a $1 bet if the digits in your number match those of the



247Chapter 9 Media Exercises

winning number in any order. You choose which of these two bets tomake. On the average over many bets, your winnings will be

mean amount won = payout amount × probability of winning

What are the mean payout amounts for these two bets? Is one of the twobets a better choice?

9.48 An edge in Pick 4. Exercise 9.45 describes Pick 4 lottery games. Somestates (New Jersey, for example) use the “pari-mutuel system” in whichthe total winnings are divided among all players who matched thewinning digits. That suggests a way to get an edge. Suppose you chooseto try to match the winning number exactly.(a) The winning number might be, for example, either 2873 or 8888.

Explain why these two outcomes have exactly the same probability.(b) It is likely that fewer people will choose one of these numbers than

the other, because it “doesn’t look random.” You prefer the lesspopular number because you will win more if fewer people share awinning number. Which of these two numbers do you prefer?

Chapter 9 MEDIA EXERCISES

9.49 What probability doesn’t say. The idea of probability is that theproportion of heads in many tosses of a balanced coin eventually getsclose to 0.5. But does the actual count of heads get close to one-half thenumber of tosses? Let’s find out. Set the “Probability of heads” in theProbability applet to 0.5 and the number of tosses to 40. You can extendthe number of tosses by clicking “Toss” again to get 40 more. Don’t click“Reset” during this exercise.

APPLET

(a) After 40 tosses, what is the proportion of heads? What is the countof heads? What is the difference between the count of heads and 20(one-half the number of tosses)?

(b) Keep going to 120 tosses. Again record the proportion and count ofheads and the difference between the count and 60 (half thenumber of tosses).

(c) Keep going. Stop at 240 tosses and again at 480 tosses to record thesame facts. Although it may take a long time, the laws of probabilitysay that the proportion of heads will always get close to 0.5 and alsothat the difference between the count of heads and half the numberof tosses will always grow without limit.

APPLET

9.50 Shaq’s free throws. The basketball player Shaquille O’Neal makesabout half of his free throws over an entire season. Use the Probabilityapplet or software to simulate 100 free throws shot independently by aplayer who has probability 0.5 of making each shot. (In most software,




the key phrase to look for is “Bernoulli trials.” This is the technical termfor independent trials with Yes/No outcomes. Our outcomes here are“Hit” and “Miss.”)

APPLET

(a) What percent of the 100 shots did he hit?(b) Examine the sequence of hits and misses. How long was the longest

run of shots made? Of shots missed? (Sequences of random outcomesoften show runs longer than our intuition thinks likely.)

9.51 Simulating an opinion poll. A recent opinion poll showed that about65% of the American public have a favorable opinion of the softwarecompany Microsoft. Suppose that this is exactly true. Choosing a personat random then has probability 0.65 of getting one who has a favorableopinion of Microsoft. Use the Probability applet or your statisticalsoftware to simulate choosing many people independently. (In mostsoftware, the key phrase to look for is “Bernoulli trials.” This is thetechnical term for independent trials with Yes/No outcomes. Ouroutcomes here are “Favorable” or not.)(a) Simulate drawing 20 people, then 80 people, then 320 people. What

proportion has a favorable opinion of Microsoft in each case? Weexpect (but because of chance variation we can’t be sure) that theproportion will be closer to 0.65 in longer runs of trials.

(b) Simulate drawing 20 people 10 times and record the percents ineach trial who have a favorable opinion of Microsoft. Then simulatedrawing 320 people 10 times and again record the 10 percents.Which set of 10 results is less variable? We expect the results of 320trials to be more predictable (less variable) than the results of 20trials. That is “long-run regularity” showing itself.

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Introducing Probability - virtual.yosemite.cc.ca.usvirtual.yosemite.cc.ca.us/jcurl/Math 134 3 s/chap09np.pdf · We might suspect that a coin has probability 0.5 of coming up heads