In grades 9-12, the mathematics curriculum should include the continued study of probability so that all students can--

• use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty;
• use simulations to estimate probabilities;
• understand the concept of a random variable;
• create and interpret discrete probability distributions;
• describe, in general terms, the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed;

and so that, in addition, college-intending students can--

• apply the concept of a random variable to generate and interpret probability distributions including binomial, uniform, normal, and chi square.

Focus

Probability provides concepts and methods for dealing with uncertainty and for interpreting predictions based on uncertainty. Probabilistic measures are used to make marketing, research, business, entertainment, and defense decisions, and the language of probability is used to communicate these results to others. In grades 9-12, students should extend their K-8 experiences with simulations and experimental probability to continue to improve their intuition. These experiences provide students with a basis of understanding from which to make informed observations about the likelihood of events, to interpret and judge the validity of statistical claims in view of the underlying probabilistic assumptions, and to build more formal concepts of theoretical probability.

Discussion

Students in grades 9-12 should understand the differences between experimental and theoretical probability. Concepts of probability, such as independent and dependent events, and their relationship to compound events and conditional probability should be taught intuitively. Formal definitions and properties should be developed only after a firm conceptual base is established so that students do not apply formulas indiscriminately when solving probability problems. At this level, the focus of instructional time should be shifted from the selection of the correct counting technique to analysis of the problem situation and design of an appropriate simulation procedure. Some students also should be encouraged to approach problems from the perspective of a theoretical model.

It is also important for students to understand the differences between, and the advantages associated with, theoretical and simulation techniques. Even more important, students should value both approaches. For example, students might obtain a result through the application of a theoretical model and validate that result through simulation. What should not be taught is that only the theoretical approach yields the "right" solution.

Although probability provides useful models for the solution of problems in fields such as medicine, physics, and economics, many of the problems in daily living also can be better understood from this perspective.

Suppose Anne tells you that under her old method of shooting free throws in basketball, her average was 60%. Using a new method of shooting, she scored 9 out of her first 10 throws. Should she conclude that the new method really is better than the old method?

This problem can be addressed within the core curriculum at increasing levels of abstraction.

All students should first identify the real (statistical) question: What are the chances of shooting at least nine out of ten if you normally shoot 60%?

Level 1: Students model the problem by associating baskets with the integers 4-9 inclusive and misses with the numbers 0-3 inclusive. They then roll a fair icosahedral die (twenty faces with the digits 0-9 appearing twice) ten times. If nine or more "baskets" occur, count the trial as a success. For the first trial (see fig. 11.1), the digits 4-9 occur only seven times, so this trial is not a success. Students repeat the experiment nine more times and determine the percentage of successes.

Fig. 11.1. Free-throw shooting simulation

For this set of ten trials, there is only one success--trial 8. So the chances of making at least nine out of ten shots is 1/10, or 10%. To obtain a better estimate of this probability, the results from the class would be pooled.

Level 2: Students use a random-number table (fig. 11.2) for the same kind of simulation. The digits 0, 1, 2, 3 correspond to misses; the digits 4, 5, 6, 7, 8, 9 to baskets. Each trial consists of selecting ten digits; a success (nine or ten baskets in ten free throws) occurs if the digits 0, 1, 2, 3 occur less than twice in each set of ten. For the trial shown, there are four misses and six baskets; hence the trial is not a success.

Fig. 11.2. Selection of ten uniformly distributed random digits

Level 3: Students develop and use a computer program with a random-number generator to run the experiment 1000 times and compute the ratio of successful trials.

REM *** This program runs 1000 10-shot experiments shooting free



throws







Successes = 0



FOR trial = 1 to 1000



   Counter = 0



   FOR shotnumber = 1 to 10



           X = INT(10*RND)



           IF X > = 4 THEN Counter = Counter + 1



   NEXT shotnumber



   IF Counter > = 9 THEN Successes = Successes + 1



NEXT trial



PRINT "The probability of shooting at least 9 baskets is";



Successes/1000







>RUN







>The probability of shooting at least nine baskets is .048.

Level 4: Students relate the problem to the binomial distribution and the specific question to the binomial probabilities (10, 9; 0.6) and (10, 10; 0.6).

The probability of success: probability of making exactly nine shots + probability of making all ten shots

On completing work at any one of the levels, students would be asked to interpret their obtained probabilities and formulate a response to Anne about her new method of shooting free throws.

The reader should note that although the problem setting considered here remains the same for all students, the instructional treatment varies not only in the content and its complexity of language and notation but also in the type of resources (polyhedral dice, random-number table, computer-based random-number generation) used. The impact of computing technology on the teaching of probability can be clearly seen by comparing the results obtained by informal simulation methods (level 3) with those obtained by formal analytic methods (level 4). Even students who successfully completed work at levels 1 or 2 could discuss and then use a computer program based on their model.

Once students have acquired an intuitive understanding of a random variable, they can extend the concept of the probability of an event to the development of a probability distribution. For example, students could associate the outcome of each roll of two dice with the product of the numbers on each face as illustrated in figure 11.3.

Fig. 11.3. Introducing the concept of a random variable

Probabilities now can be assigned to each outcome, experimentally by simulation or theoretically by constructing a table of outcomes. This can be expressed by college-intending students in formal notation, such as P(x = 6) = 1/9, where x denotes the random variable. All students can construct a graph of this discrete probability distribution to answer such questions as, What is the probability of obtaining a product greater than 10?

In addition to generating discrete probability distributions, such as the binomial (n, x; p) for various values of n and p, college-intending students could use a mathematical statement for a random variable (e.g., X = INT(- 5*log(RND)), where INT is the greatest-integer function and RND represents a random number, 0 < RND < 1, and construct the associated frequency distribution (in this example, Poisson). This kind of distribution occurs in such situations as queues and traffic flow.

Furthermore, college-intending students could also relate the coefficients of the terms in the expansion of a power of a binomial expression--for example, (p + q)ⁿ--to the frequency of events in a binomial distribution. Finite random walks on a number line or in the first quadrant are appropriate applications of these concepts and can easily be simulated on a computer.

Because the distributions of data from many real-world phenomena can be closely approximated by the normal curve, all students should become familiar with the geometric properties of its graph and should be able to use either probability tables associated with the curve or computer software to solve problems. Here is one example:

Assuming that gas consumption for a car model is normally distributed with a mean of 25.7 mpg and a standard deviation of 2.9 mpg, how likely would it be that a particular car of this model gets at least 30 mpg?

To solve this problem, it is first necessary to standardize the random variable (gas consumption) using transformations on the mean and standard deviation as discussed in the standard on statistics.

Here P(mpg 30) = P(z 1.48). In figure 11.4, the area of the shaded region corresponds to the probability of this event. From a standardized normal-curve table, this probability is found to be approximately .07. College-intending students should also investigate similarities between the binomial and normal distributions.

Fig. 11.4. Gas mileage distribution

Probability and statistics should be developed in a manner that highlights their interrelatedness. If students are to test hypotheses in statistics, they need a good understanding of probability distributions. As an illustration, consider the following problem:

In a taste-test experiment involving three types of cola, 30 people make a selection: 15 choose Brand X, 8 Brand Y, and 7 Brand Z. The manufacturer of Brand X claims that these data show its superiority over the others. Is his claim reasonable? How likely is it that the outcome could have occurred by chance if there were no preference?

In solving this problem, students can develop some new mathematics and in the process gain understanding of the chi-square statistic. Here, in capsule form, is one process they might follow:

1. Students agree that a 10-10-10 distribution of choices would result if there were no preference.
2. Deviations from this result can be handled so that they won't cancel each other out, by computing (10 - 15)², (10 - 8)², and (10 - 7)², as in a variance calculation.
3. In order to scale these values so that they may be compared with similar distributions, they each are divided by the expected value and then summed. The resulting value is 3.8, which is a particular value of a random variable called a chi-square statistic and is denoted ².
4. The problem now becomes finding P( ² 3.8). Students may explore this probability through simulation of the cola experiment assuming no preference and tabulating the resulting frequency distribution of this random variable, ², as in the following stem-and-leaf table for sixty trials (fig. 11.5). In this table, there are eight values for which ² 3.8; thus, the associated P(² 3.8) = 8/60, or .13+. Thus, the 15-8-7 distribution (or worse) would have occurred about 13% of the time even if there were no preference among the colas in a sample of thirty people.

   

   Fig. 11.5. Stem-and-leaf table for simulated cola experiments

5. Subsequent discussion of this result should convince students that a claim of preference for Brand X is not very strong.

In addition to providing information related to a specific (statistics) problem, these results also may be used to confirm the theoretical value to be found in a table for the chi-square statistic. This latter activity falls clearly within the realm of probability.