Chapter 6

PROBABILITY

 Not only is probability of great importance in testing, it is of great importance in all areas of human thought.  I believe its importance compares favorably with that of reading.

When we say in English that an event will probably occur, we usually mean something like:

We can specify degrees of probability without recourse to numbers by saying that something is very probable, extremely probable or nearly certain.  Such phrases suggest getting closer and closer to something and that something can be labeled nicely as “certainty.” It has turned out to be convenient to use numbers for degrees of probability, though.  Since there are as many points as we could ever need between the number 0 and the number 1, and since we seem to be approaching a particular point as something gets more and more probable, 1 serves as the point of certainty and 0 as the point of impossibility.

When we say that something is very, very probable, that it is quite likely to occur, we are trying to convey the idea of “a little looseness.”  There doesn’t seem to be any looseness at all in the connection between my getting married and my becoming a husband.  Anybody who said it was probable that I should become a husband if I marry would be playing with language--that is, kidding--or misunderstanding what the words mean.  The event of becoming a husband isn’t probable under the hypothesis of my getting married; it is absolutely certain.  Here, then, there is no looseness at all.  But absolutely certain events are rare and usually the genuinely certain ones aren’t the events in this world that we are interested in.  We are interested in contingent--chancy--events like winning a game or an election, amazing a certain person or, as teachers, having our students succeed.  We know there will be a little looseness.  What we are interested in is:  how much?  We can learn to express little bits of looseness in words, common fractions or decimal fractions.

An event that is very highly probable but not certain is one that isn’t defined certainly.  Suppose I were to have a wrestling match with the world heavyweight champion.  If everyone agrees to call the match a victory for him no matter what happens during the match itself, we might as well not bother to wrestle.  The result is a foregone conclusion.  However, by the usual rules, there does seem to be a microscopic possibility that I--smaller, older, less fit and less knowledgeable--would win.  We sometimes refer to thought and say, “It is conceivable.  I can imagine, under very weird circumstances, a victory for Kirby.”  Maybe my appearance would reduce the champion to laughing helplessness.  Maybe I would be lucky.  And here we come to the common fraction idea--if somehow, I could wrestle the champion 100 or 1,000,000 times, maybe, just maybe, I could win once.  So we say the probability is 1/100 or 1/1,000,000.  Some people would prefer to say the odds are 1 to 99 or 1 to 999,999 in my favor.  In the same way, the probability that the champ will win is 99/100 or 999,999/1,000,000.  Or the odds are 99 to 1 against me.  Or there is a 99% chance that I’ll lose.

If the probability that I will win is 1/100 and something makes it a little less likely, that something has loosened the connection between my trying and my succeeding a little bit.  The new probability might be 1/101 or 1/102.  That is, I might have to wrestle him a few more times before I could expect one measly victory.  However, handling numbers like 1/100 and 1/102 is awkward.  In fact, it is a big job to find how much the probability fell when it went from 1/101 to 1/102.  So, decimals come in.  Decimal fractions aren’t as pictorial as common fractions but they are much easier to calculate with.

Probability is an amazing subject in many ways.  Why the ancients didn’t develop it is certainly a mystery.  The first person usually cited as a probabilist is the “gambling scholar” of Italy, Cardano (1501-1576) but the real grandfathers of probability are the French mathematicians, Pascal and Fermat.  In the 1600’s, these men discussed, by correspondence, questions concerning the probable outcome of games.  However, it has only been in this century that the subject has expanded to its present incredible extent and variety.  It is certainly a mathematical subject and the purely mathematical writings on such things as stochastic processes are steadily expanding.  Probability is also a philosophical subject and such modern philosophers as Lucas, Kyburg, Salmon, Ayers and Wisdom find it worthwhile.  These men are probably (there’s that notion again) not familiar to you but if you have an opportunity to study areas of philosophy dealing with the ultimate difference between men and machines or the meaning of scientific investigation, you would meet them.  However, probability would be a limited and rarified subject if it were of interest only to mathematicians and philosophers.  Today, sciences from the physics of the tiniest bits of matter to the study of mass psychology find probabilistic concepts and calculations the only language and tools useful in many circumstances.  There is more.  Virtually every branch of government, from urban planning to crime control to weather prediction to national defense is steadily expanding its use of probability.  Business methods of planning and decision-making based on probability are continuously moving from theoretical studies in graduate schools to practical applications in on-going companies with immediate problems.  In education, the most common explicit use of probability is in graduate or research investigations.  But the concepts and some methods of manipulating probabilities are spreading.  There has been an interest in making probabilistic statements in counseling and career planning for some time.  A student might be told that he has only a 10% chance of success as a forester, instead of being told he won’t make a successful forester.  Of course, testing draws heavily on probabilistic ideas and methods.  We often make use of probability to try to differentiate between skill and luck, as in the chapter on the lady-testing-tea test.

 I must admit, though, that all uses of probability on real-world, or empirical, data result in nothing more than probabilistic statements in the end.  There is no way we can start with probabilities and merely calculate our way to certainly.  Still probabilities can give us very helpful guidance, especially after we have gained experience in reading and using them.  A famous example of the unusual insight we can sometimes get from probability but not from our personal judgment is called “the birthday problem.”  The question is:  How many people must be in a room before there is a good chance that at least two of them will have the same day of the year as their birthday?  Answers to the question are often wildly off the mark.  Many mathematics books, such as the excellent “Introduction to Finite Mathematics” by Kemeny, Snell and Thompson give the answer and a full explanation.  (There is 70% chance with 30 people and 99% with 60 people.)

If you grant that the subject may be worthwhile, what constitutes a start in learning it?  The most basic things have already been mentioned:

 Beyond these simple principles, we are already beginning to climb into the subject proper.  Most initial instruction goes from these principles into notation and manipulation.  For the use of probability in studying problems of testing and grading, our notational needs are simple.  We will write “the probability of A”, where A is some event, so often that it is handy to have the usual symbolism

            P (the event A) = P(A) =

for “the probability of the event A equals”. Occasionally we wish to be able to state restrictions or special conditions for the event.  For instance, the probability of the event “having a girl baby” could be symbolized P(having a girl baby).  If we were interested in the probability subject to the condition that the expectant mother had red hair, the typical symbolism is

P (having a girl baby|a red-headed mother).

 The vertical stroke is read “given.”  Events and their conditions may be symbolized by letters but it is important to remember that such letters stand for English phrases, not for numbers.  So, using G for “having a girl baby” and R for “red-headed mother,” we could write more quickly  

            P (G|R)

 Instruction in the manipulation of basic probabilities can be quite complex and most of the usual matters are outside the scope of this book.  However, one part of the traditional material is useful and easily mastered.  It concerns successful repetitions of an event.  Suppose there is a 50% chance of having a girl baby and a 50% chance of having a boy.  The question is:  What happens to the 50% as we specify longer and longer strings of girls born to the same mother?  Some students have had enough instruction in probability to remember something about chances remaining the same.  Partial memories will lead them astray if they believe that the chances of two girls (and no boys) in a family of 2 children are the same as the chances of one girl in a family of one child.  There is something that remains the same and it is just that initial 50%.  We get the probability of an event happening “in a row” by using the base probability, the 50%.  But whatever we do, we clearly must arrive at smaller chances for more difficult or more unlikely events.  It is harder to get two girls in a row, with no intervention from boys, than to get a single girl at the beginning.  It is harder to get five heads in a row with a coin than to get one head with one flip.

If half the babies born are girls, and all those with daughters have a second child and the chances of a girl are always 50%, then half of those with a girl the first time will have a girl the second time.  Half had a girl the first time and half of that half will have a girl the second time.

            P (G twice) = P(G) x P(G) =

            [P (G)]2 = 1/2 x 1/2 = 1/4 = 25%

How about three daughters?  You can’t have three daughters if your first two children aren’t girls.  You have a 1/4 chance [P(G)]2 of two girls.  If you are in the two girls group, you have that same old 50% chance of another girl.

            P (G three times) =
            P(G) x P(G) x P(G) =
            [P(G)]3 = 1/2 x 1/2 x 1/2 =
            1/8 =12.5%.

So the probability of an event occurring N times in a row equals N of the base probabilities multiplied together.

            P (event N times in a row) =  [P (event)]N

The next step in probability instruction is usually to learn how to calculate the chance of specified combinations of successes and failures.  We’ll skip that.  

That about winds up a basic introduction to probability for testing and grading use.  The only important matter omitted so far is the actual obtaining of a probability.  In purely mathematical work, this step may be omitted as being a problem of nonmathematical practical interest.  However, we will need probabilities.  How the initial base probability is best obtained is the subject of sharp and continuing debate.  There are four main methods--counting, direct opinion, analysis of bets and mathematical models.

For the present, we will restrict ourselves to counting.  How do we obtain the probability of randomly selecting a girl from a class?  We count the number of girls  and the number of nongirls (sometimes called “boys”). Then the probability is

            number of girls
            number of students in all

This “frequentist” definition of a probability can be applied to many situations.  When I ask students about the probability of randomly selecting a person from the class who  is mentally healthy, some of them protest.  Mental health is a vague concept, they say, and the subject of disagreement and debate, even among experts.  I agree, but the probability can still be defined in the same way:

P (randomly selecting a mentally healthy person) =

            number of mentally healthy student
            total number of students

Is this idea a vacuous one?  No, because various definitions could be applied to a group to see what sort of probabilities emerged.  In such an investigation, new insights into the meaning of mental health and its appropriate definition might be uncovered.  Why might a teacher want the probability of selecting students with various characteristics?  Because probabilities offer a better grasp of the complexities of a class than mere percentages.  Some people say that being told 80% of a class is right-handed something conveys little to them.  But thinking of the 80% as the chance of selecting a student can help.  In the 80% right-handed class, there is a 64% chance of picking a right-handed student twice in a row.  It is just about uncertain whether you can choose three such students in a row or not.  (P) = .8 x .8 x .8 = .512)  However, it is definitely unlikely that you will pick four in a row.  (P) = (.8)4 = .41)  The more you know about probability, the more you can play with that 80% figure.  Probability can clearly make percentage figures more meaningful and useful.

References

Feller, W., An Introduction to Probability Theory and Its Applications, 3rd edition, 1968, Wiley.

Kemeny, J. et al., Introduction to Finite Mathematics, 3rd edition, 1974, Prentice-Hall.

Exercises

Read through an almanac and convert some statistics from it into probabilities.