Topics:

Introduction   Accuracy, Precision and Uncertainties     Probability      Finding the Perfect Mate

Probabilities in Science        Summary        Questions        Bibliography
 
 

         There is a story of a physicist who had a horseshoe hanging above the door of his laboratory.  Knowing that this physicist’s prime law was “there ain’t no such thing as Luck, they asked him if he really thought it would bring luck to his experiments, he replied: “No, I don’t believe in superstitions.  But I have been told that it works even if you don’t believe in it.”  … Does he believe in it?

Introduction

        Any attempt to describe our physical world must begin with a consideration of the measurement process.  In the last unit we investigated the nature of measurement and functional relations.  Now, we look more deeply into what the measurements mean, how they are reported, and particularly where there is question as to the validity of some value, attempt to apply probability theory to the process.  We do not intend to thoroughly  develop the probability theory, nor the extensive kinds of statistics commonly used in science, but rather provide the reader with a sense of comfort in either accepting or rejecting ideas and a knowledge that certain results do indeed have a reasonable basis.  A great number of different phenomena can be describe only in terms of probability, such as scattering events, radioactive decay, half-life, even a person’s own life expectancy.  Too often human beings want answers that are exact and this is not necessarily how the world works.  Probability affects much of our real world.  We will see over and over again, how, in the very small domain of atoms that probability governs that level.  Even the world we daily interact with depends more on statistical averages of certain quantities.  As a final investigation we look into the nature of risk, and as a matter of interest, without being judgmental about ethical and related issues, read a journal article on the comparison of risks with radioactive and non-radioactive waste products as health hazards.

Accuracy , Precision and Uncertainties

 When measurements are made and values given, we should immediately recognize that they are never exact.  All of the fundamental constants in the universe, the acceleration due to gravity, the speed of light, the mass of an electron, are all known only to a certain degree, perhaps eight or nine places right of the decimal. Each quantity, each measurement, is limited by two factors, called precision and accuracy.  Accuracy is concerned with the closeness of the result to the true value.  Consider an example of a marksman shooting a target.  How close the bullet holes are to the bullseye indicates the accuracy.  On the otherhand, precision describes the consistency of the rifle and the shooter, or how smll a circle that can be drawn about the group of shots.  Numerically, precision refers to the number of significant figures in an answer.  Accuracy tells us how close to describing the quantity the measurement really is.  A value can in fact be very precise (many decimal places for example) but very inaccurate.

         Techniques for making actual measurements of physical quantities are discussed in the laboratory.  They depend upon the instrumentation and the environment as well as human limitations.  Whenever they are made, however, they must be done so with an indication of how accurate the result is.  This is frequently called the uncertainty.  Some authors call it the error in a value.  Since error implies something wrong in the process, we prefer to use the former term, uncertainty.

         Keep in mind that the measurement process is vital to the scientific method.  Collection of data, proper interpretation of that data and using it to confirm or formulate a model is necessary. Sometimes there is an opportunity to measure a quantity repeatedly without constraints.  Other times, such as determining quantities of the corona during a solar eclipse, will be a once in a lifetime chance.  In this case the uncertainty must by estimated (with great care to be conservative) from the instrumentation and the environment.  In other cases, where there are continuous values to be repeatedly measured in many trials of an experiment, the uncertainty and the value can be determined directly through the statistical process.

        In the laboratory, we report measured values with their respective uncertainties.  For example, the width of a room might be

        This basically means the width of the room is somewhere between 3.95 and 4.05 metres. When many trials are made of an experiment and many measurements of the appropriate variables are available, a statistical average describes the variable quite well.  The expected outcome of the experiment is described by the mean value.  The Arithmetic Mean is found directly by adding the results of individual measurements and dividing by the total number of measurements:

Where <X> indicates the mean value of the measurements and X₁ refers to the value of the variable X on the first trial, X₂ tells us the value of the variable X on the second trial, and so forth.  One must be careful not to confuse this process with other measurements of central tendency.  There is also a geometric mean, etc. The reader is advised to investigate these other measures of central tendency in any elementary statistics book. It is not the intent here to help the learner know the various kinds, but rather know that they do exist.

        Just as measured values have uncertainties associated with them, calculated values, which depend in part on measured and hence uncertain values, also have an uncertainty associated with them. Such determinations of the uncertainties are covered in the laboratory section of this course. The more measurements of a value, the smaller the uncertainty.

        There is a theorem in mathematics which tells us that if we make an infinite number of measurements, the arithmetic mean is in fact the true value of the quantity we are interested in. It is called Bernoulli's Theorem. Of course, we can't make an infinite number of measurements of anything, but frequently we can make a large number of measurements. Sometimes ten or twenty measurements of the quantity, under, controlled conditions, is sufficient to determine the mean value.

        The accompanying uncertainty or variation, or the range that the true value lies in around the mean, can be calculated directly through a statistical process. Typically this is performed on a computer and the reader need not be concerned with the mathematical process used. Instead, the reader should recognize that the process is well defined and the results can be stated clearly describing a mean value, or expected outcome (prediction from our model) with a confident range of uncertainty or degree of precision. In other words, these values do have a logical basis for their existence.

    A simple example of this might be finding the average class score on a particular test. The Arithametic Mean is defined by adding up the total scores and dividing by the number of tests given. The arithmetic mean typically describes reasonably well how the class did. Some 'students actually did better. Some did worse. And there is a range of uncertainty about that value. Teachers frequently use a quantity called the standard deviation, giving a range of uncertainty about the mean. Small percentages of students will achieve scores two or more standard deviations above and below the mean. It can be shown that we can expect 68% of the scores to be within one standard deviation of the mean and only 5% above or below two standard deviations. This works for a large sample, not for a small class. Such a distribution of scores is sometimes called a "bell" curve. It is symmetrical about the middle. Suppose we have a cup of popcorn in a pan on the fire. Let us measure the time it takes for each kernel of corn to pop. Typically we place the pan on the fire and nothing happens for a while. Suddenly one explodes and we get excited. Then another, and then two more. Pretty soon it seems like they are all popping at once. After a time, the rate slows down and only a few are poping. Now only two, and then one, and we have to wait a long time for the last one. Some never do pops That distribution of results will also give us a bell curve.
Figure 3.1 Average of a large sample. Note the standard deviations, one measure of variation about the mean value.
 
 

         The concept of average should not be taken lightly for the results of measurements of continuous variables typically describe averages. Consider the molecules in the room as a gas of random velocities. It will be shown later that in fact the temperature of the gas characterizes the average kinetic energy of those molecules. This means that for a specific temperature, the average velocity is well known by a mathematical model. This does not mean to imply that all the particles of the gas move at the same speed. Rather, this value is an average , determined through some mathematical procedure. Any specific particle may actually be moving somewhat faster or somewhat slower than the average. Later too, we will see that the basis for description of many parameters in the quantum world do relate to probabilities and probable values.

The Basics of Probability

In some cases there are several possible outcomes to an event. For example, suppose we throw a die with six faces. There are six possible outcomes, and without pretending to be experts in the nature of chance, we should recognize that if the die is fair (that is, it is honest and each face has an equal and independent chance of turning up) we expect that the probability of throwing a particular face is 1 out of 6. We would expect that each face would turn up once in six throws. A real experiemnt where we could throw a die six times might not yield those results, but after a large number of trials, we can expect to see that each face, on the average, showed up 1/6th of the time.
 
 

        If we want to determine a quantity that depends on several independent events, the result is merely the sum of the probability for the outcome of each event. Consider our die again. What is the probability of throwing an ODD number? Well, there are three ways to do this, a 1, a 3, or a 5. The probability for each independent event is identical, or 1/6th. Adding these we can show that the probability of throwing an odd number is 1/6 + 1/6 + 1/6 or 1/2.

        If the outcome of an event depends in fact on a sequence of several events that depend on each other, then the probability for that result is expectedly the product of the individual probabilities. Suppose we have a bottle with equal numbers of black and white spheres in it. The probability of drawing a white sphere is 1/2. Now, take that white ball and place it back in the bottle. The probability of drawing a white ball is again 1/2. But the probability of drawing two white balls in a row, a sequence of events is 1/2 X 1/2 or 1/4.

        It is important to recognize that these two methods of calculating probabilities are the main techniques used in problems. One depends on the outcomes of mutually independent events and the other on independent, sequential happenings. Frequently, a combination of the two techniques is used. Let us look atthe values of the outcomes of a pair of dice. Let X represent the sum of the two dice and P(X) be the probability of that sum ocurring. The reader is invited to complete the following table:
 
 
 

         We should recognize that there is no way to throw a 0 or one and only one way to throw snake-eyes or 2. Both dice would have to be ls. There is also only one way to throw a 12. Both dice would have to be 6s. But there are two ways to throw a 3. The first die could be a 1 and the second a 2 or, the first could be a 2 and the second a 1. Similarly for the other values. Note that 7 can be thrown in more ways than other combinations. What is the most probable outcome of a throw of a pair of dice? A look at Table 3.1 tells us that a 7 is most likely since there are six ways to achieve this outcome.

Finding the Perfect Mate

        In our lives we sometimes are "looking" for something that may in fact be difficult, if not impossible to achieve. Perhaps you have heard the story of the fellow that was looking for the perfect woman. When he found her, he was disappointed because she was also looking for the perfect man (which he was not). There is a moral to the story, but from a practical point of view, have you ever wondered about the existence of a perfect mate? Let us attempt to estimate the probability of finding that elusive being.

        We must realize that our picture of the perfect mate depends on a combination of factors. Assuming that few of us would have the financial means to search world wide for this person, let us restrict our exporation to the United States of America. Extending this to the rest of the Earth or even the entire galaxy is merely mathematical formalism. You can do it for yourself. (Remember too, that we will look for the ideal mate, not just an acceptable one.) Let us list some factors that might affect our decision. You may add more if you do your own calculation.
 

        Sex, Race, Religion, Political Preference, Height, Physique, Complexion, Facial Features, Health, Financial Status, Intelligence, and Personality are some of the factors we choose to consider. While some of the above may not matter to you we include them merely as an intellectual exercise.

         The population of the United States is about 290 million. (We'll see what the 2000 Census says when the info is compiled.) Let us say that roughly half are of the appropriate sex. (Small perturbations to a 50/50 ratio will not affect the result.) That means there are 145 million possible indiviuals to select from. Let us also suggest that half of them are of an acceptable race. While this value is subject to discussion, the result will hardly be affected. That leaves us about 72 million, still a large number to search.

        Next, let us suppose that 1/4th are of the preferred religion. For some persons this is a very critical factor and for others, perhaps less important. The 1/4th factor leaves us 18 million possibilities, still a large number. Suppose that half are of the right political perspective. Again, unless the perfect mate must be of an extreme minority, a one-half probability for this factor will give reasonable results. That leaves us about 9 million possibilities. For most people of average height, half of the population is of the same height or taller. For some persons this figure may necessarily be different; for example, a six foot three inch tall woman who wants a much taller man may find a smaller percentage is more appropriate. Using one-half again (admittedly the fraction may be significantly smaller but the process is what is important), we get 4 1/2 million remaining, still a lot!

        Suppose now that as many as 1/4th have the right physique, 1/4th have the right face, and 1/4th have the right complexion. We have been a bit fussier with these characteristics, but the average person may be even more so. This now reduces the number remaining to about 70,312. Let us go on to suggest that 1/4th are in perfect health, but that only about 1/10th are financially "perfect." Now, in the entire United States there are 1757 possibilities.

        Let us say that the intelligence should be at least one standard deviation above the norm (about 16% or one of six.) We've now reduced the possibilities to 293. If one of four have the right personality (surely one would be more selective than this) so there are 73 perfect mates in the USA. Say that half of them are of the right age range and we have about 36 ideal mates. That's less than one per state; of course it is expected that the reader has found or will find the ideal mate.

        Recognize that if you add in factors such as likes and dislikes, do they do dishes and windows, are they sensitive, caring, like kids, etc., the probabilities will become extraordinarily small.

         What we have done with this simple example is to apply one of the two types of probability conditions, namely equation (3-4), where the outcome of a situation depends sequentially on several conditions. It is the product of all the probabilities. In this case we multiplied the resultant probability by the total number of people available and found how many perfect or ideal mates exist. There are more complex kinds of probabilites that can be calculated, but that is not our intention here. We only intend to show how probailities in essence can be used to describe real world phenomena.

        We can use probability to describe phenomena beyond our normal real world experiences too. Astronomers use the Drake Equation, devised by Frank Drake, to determine the number of civilizations. Like our search for the ideal mate, they attach estimated probabilities to the number of stars in the galaxy, the fraction of those that "live" long enough for intelligent life to develop, the average number of planets per star, the fraction of these that are suitable for life to develop, etc. The product of such probabilities multiplied by the total number of stars in the galaxy gives us the number of intelligent civilizations we might be able to communicate with. Depending on the assigned fractional probabilities, the results can range widely. It is not our intention to describe the results and debate them, but rather to help thew reader to better understand how estimates are made in science, and what they mean. Many quantities are merely probabilitistic estimates and not firm results. See Question 10 at the end of this chapter.

Using Probabilities in Science

        Probabilities can be used to: 1) make decisions and 2) to reach conclusions in science. It should be recognized that decisions made on the basis of probability have a finite chance to be the wrong choice. Conclusions reached may also be wrong. In any cases, we do attempt to eliminate errors. It should be recognized that good decision makers typically (although they do not formally calculate probabilites) estimate relative chances of success based on their experiences.

         Consider the following example of using probabilites to make decisions. Suppose we have two bottles with white and black marbles in them. Both bottles have the same number of marbles in them. Bottle # 1 has 50% white and 50% black marbles while Bottle # 2 has 75% white and 25% black ones. Now, let us say that we left the covers off the bottles so that we lost any way to identify which was Bottle # 1 or Bottle # 2. Let us draw a marble from one of them. Check to see if it is white and replace it and draw again. Do this 10 times and suppose the outcome is that for all 10 draws a white marble results. Two possible conclusions can be drawn. Either this is Bottle # 1 or it is Bottle # 2. Now, intuition might tell us that this must be Bottle # 2. Let's see. These are successive independent events so the total probability is the product of the probabilities for each event.

For Bottle # 1:

For Bottle # 2:

i.e. the probability that 10 whites in succession were drawn from Bottle # 1 is 57 times more likely than for drawing 10 whites marbles from Bottle # 2. Our intuition is correct: Our decision would be that this must be bottle # 1. Of course we might be wrong, but probably not.

        The reader will not be asked to make formal calculations of such probabilities, but rather to recognize that their use results in credible decisions.

        Reaching conclusions in science is a very interesting process that few really understand. It seems that many science books lay out the scientific method as a fool proof procedure by which one starts from simple facts and develops a grand theory. Nothing could be farther from the truth. In fact, just as a thought process, this sounds more like the Greek method. Actually, some of the greatest ideas were developed thanks to some intuitive leap from the known to the unkown, based on certain experiences. Einstein could never have developed the notion of the equivalence of gravitation and accelerating inertial reference systems in a purely logical manner, let alone applying it to our physical world.

        Still, probability plays an important role in reaching conclusions in the scientific method and it gives us a sound basis for acceptance or rejection of a hypothesis. The scientific method itself consists of:
 

1.  State the Problem - we don't want vaguenss of focus, but rather a clear, precise formulation of the phenomena.
2.  Formulate Hypothesis - it must be simple, yet comprehensive.
3.  Design an appropriate experiment to test the hypothesis. This includes data collection, etc.
4.  Collect Data - avoid bias, which more than likely is present no matter how much we try to avoid it.
5.  Interpret the Data - analytically apply appropriate statistics including probabilities and tests as designed for the hypothesis.
6.  Draw conlusions - Accept or reject the hypothesis. This ultimately can lead to the establishment of theory.

        Hypotheses are typically stated in two ways, dependent upon the design of the experiment. The following is not meant to cover the many different ways experiments can be designed and performed, but rather to show a couple of typical ones. There are many varied statistical tests too, not just the ones we discuss in class or herein. What is important is that we understand that there must be a reasonable basis for decision making and drawing conclusions. While hunches and intuition may provide an initial direction, only solidly based statistical tests can give us results we can rely on. You especially would like it to be so if you were betting your paycheck on it, or if a medical procedure that could save your life was selected from a choice of several alternatives.
Suppose we want to play a game of flipping a coin, but we will only play if the coin is honest i.e., there is an equal chance for each side, heads or tails, to be the outcome of the flip. Our hypothesis is that the coin is fair or honest if the probability of heads (or tails) is .50.

We can do a quick experiment that tests this value so we will either accept or reject the hypothesis. Gamblers do this with dice and coins rather quickly. Another way to do this is to compare two means. Suppose the two means are X and Y. One way to state this is by the hypothesis that X=Y or:

      Usually this is stated in a negative way such that  there   is no (statistical) difference between the two values. This is called the null hypothesis. The statistical tests for this are stronger to reject the null hypothesis meaning that our    hunch  is  to accept the normal hypothesis.)

        Sometimes we don't understand the mechanism by which a certain phenomena works, but we see strong links. Just because a link exists, does not necessarily mean that there is a cause and effect relationship. For example, we could do a statitistical analysis of IQs and shoe size. There will definitely be some correlation between the data. It might be strong or it might be weak. There is a statistical test, a correlation coefficient which ranges from -1 to +1. -1 indicates negative correlation, 0, essentially no correlation and +1 positive perfect correlation. One should be careful not jump to conclusions.

        Recently a survey of all National Merit Scholars attempted to find what was common among them. They were widely distributed about the nation, came from varying ethnic backgrounds, worshipped at different churches, etc. But there was one question that they all answered yes. (There really sems to be no reason the question was included in the survey but the result is interesting.) All of the National Merit Finalists sat down and ate the evening meal with their families. Now, it would be ridiculous to say that eating the evening meal with your family will make a student a National Merit Finalist. It overlooks obvious intellectual differences. Hut still the result is somewhat suggestive that strong family ties and certain values may be supportive in developing academic talent. One must be careful.

        Not too many years ago a study was published in the New England Journal of Medicine indicating that persons who consumed water softened by an ion exchange home water softener showed higher incidences of heart problems. The public reaction to this was to try to avoid drinking soft water. The conclusions were drawn based on mere correlation. Some time later the same researcher also found that those who consumed soft water were also more likely to have automobile accidents. Correlation does not imply causation!

        Cigarette smokers have significantly higher incidences of emphasema, heart problems, lung cancer, etc. The actual mechanisms by which, for example, smoking causes cancer or whatever are not well understood. Read the ads in public newspapers and magazines. The tobacco industry continues to reject the notion that tobacco causes these problems. They state that no mechanism has been conclusivley shown as a cause and that correlation is merely chance. They are right, of course, but use your head and good judgement. In this case the correlation is so strong compared to non-smoking groups that one must admit to the dangers of cigarette smoking.
 

For a given hypothesis, our statistical test may make it possible for us to commit  two kinds of errors:
 

1. Type I error - that in fact we reject H when it is true
2. Type II error - that we accept H when it is false

        Suppose a new electronic device is discovered that claims to save money and increase productivity. Let X be the new and Y the old values. We don't want to accept H: X > Y  if it in fact were false.

        Generally, Type I errors will result in less serious consequences. Of course, we try to make our decisions at a reasonable level of confidence so we rarely make those errors at all.
 

        In 1987 the President nominated Judge Bork to the Supreme Court. This was indeed a controversial nomination and there probably was some partison opposition. Yet many reasonable thinking persons found it better to reject the nomination even if his questionable background was okay rather than to accept him when in fact he could have been damaging to the system. This is a clear example of how the hypothesis testing is used in a generic sense. Rather than developing detailed mathematical tests and table of probabilities to make the decision, the decision maker's experience formed the basis for this.

         In the following chapters we will not have the opportunity to apply the decision making process. We looked at it here not to confuse the reader, but rather to inform him/her as to how certain decisions are made and conclusions reached scientifically. Frequently the FDA has been criticized for not permitting certain drugs to be put on the market. Their decisions typically involve acceptance or rejection of a hypothesis. Remember that a Type I error is less serious. Think of the ramifications if a drug that were in fact harmful rather than helpful, were permitted to be marketed.

         The process of using probabilities to make decisions or reaching conclusions is actually applioed well by successful gamblers or successful business persons. They rarely directly calculate appropriate probabilities and statistics but their prior experiences give them almost instinctive abilities to recognize correct paths to follow. Think about how these techniques apply to the business world.

Summary

        Statistics are surely mathematical in nature, but are necessary tools to describe our physical world. Arithmetic means and standard deviations, with a measureable degree of accuracy and precision, describe objects around us. We have moved beyond the primitive- usage of a "bunch" or "lots" to describe amounts and instead use power of ten notation for multiples or fractions of base units. All quantities are described with uncertainties indicated. Even the most funbdamental quantities, such as the speed of light or the mass of an electron, are known only so well (precision to a limited number of decimal places.) A large sample of values results in a "bell" shaped distribution where the mean and standard deviation are well defined.

        Probabilities describe outcomes of experiments that depend on chance. Much of the world is in fact dependent upon probability. Assuming equal and independent , unbiased distributions, two general techniques are used. Those probabilities whose outcomes depend upon independent events is the sum of the individual probabilities. Those outcomes dependent upon sequential events or combinations of simultaneous conditions is the product of the individual probabilities.

        Probabilities can be used to make decisions by comparing probabilistic outcomes of several choices of systems. Reaching conclusions requires acceptance or rejection of a hypothesis. Most statements tested statistically are stated in the form of a null hypothesis.

        There is a finite chance that in fact we would make an error in our decision. We usually, via statistical methods, set this at less than 5$. It is far less serious to make a Type I error than a Type II error.

        Risk analysis involves a blend of probability determinations, decison making, and reaching conclusions.
 
 

QUESTIONS
 

1. Distinguish between Accuracy and Precision.
 

2. Give some examples of phenomena described by standard distributions or the bell curve. (Example: IQs)
 

3. Compare the probabilities of an outcome depending on mutually independent events with those dependent upon sequential or dependent happenings.
 

4. From Table 3-1, what is the probability of throwing a 9 with two dice? Describe the combinations that would give this result.
 

5. Calculate the probability of finding your ideal mate.
 

6. Determine the number of "perfect" clothing stores in the United States. Consider such factors as number of clothing stores per city, location, decor, salespersons, prices, kinds of clothes, etc.
 

7. Compare Type I and Type II errors. Give examples of each.
 

8. Give an example of a business setting where a manager has made a Type I or Type II error.
 

9. Read Risk Analysis of Buried Wastes from Electricity Generation by Bernard Cohen (See Bibliography.) Discuss the various kinds of probabilistic determinations made regarding decisions and conclusions drawn. What can you conclude?
 

10. Visit the web site:  The Drake Equation.  Discuss the probabilities of extraterrestial beings finding life on this Earth, and then of visiting it.
 

Bibliography
 

1. Blakeslee, David W. and Chinn, William G, Introductory Statistics and Probability, Houghton Mifflin, Boston, 1997 .

2. Jacobs, Harold R., Mathematics - A Human Endeavor, W.H. Freeman Co., San Francisco, 1995.

3. Dowdy,  Shirley, and Wearden, Stanley, Statistics for Research, John Wiley  & Sons,New York, 1983.

4. Rozanov, Y.A. , Introductory Probability Theory, Prentice-Hall, Inc.,  Englewood s, New Jersey, 1969.

5. Goldsmith, Donald ands Owen, Tobias, The Search for Life in the Universe, Benjamin Cummings Co., New York, 1980.

6. Cohen, Bernard L., "Risk Analysis of Buried Wastes from Electricity Generation", American Journal of Physics, Vol 54, No 1, January 1986. ''

7. Hacking, Ian, "Trial by Number", Science, November 1984.

8. Challenges for Tomorrow-Pers actives From Space, a NASA Conference, Orlando, Florida, 1987.