In grades 5-8, the mathematics curriculum should include explorations of probability in real-world situations so that students can--

• model situations by devising and carrying out experiments or simulations to determine probabilities;
• model situations by constructing a sample space to determine probabilities;
• appreciate the power of using a probability model by comparing experimental results with mathematical expectations;
• make predictions that are based on experimental or theoretical probabilities;
• develop an appreciation for the pervasive use of probability in the real world.

Focus

Probability theory is the underpinning of the modern world. Current research in both the physical and social sciences cannot be understood without it. Today's politics, tomorrow's weather report and next week's satellites all depend on it. (Huff and Geise 1959)

An understanding of probability and the related area of statistics is essential to being an informed citizen. Often we read statements such as, "There is a 20 percent chance of rain or snow today." "The odds are three to two that the Cats will win the championship." "The probability of winning the grand prize in the state lottery is 1 in 7 240 000." Students in the middle grades have an intense interest in the notions of fairness and the chances of winning games. The study of probability develops concepts and methods for investigating such situations. These methods allow students to make predictions when uncertainty exists and to make sense of claims that they see and hear.

The study of probability in grades 5-8 should not focus on developing formulas or computing the likelihood of events pictured in texts. Students should actively explore situations by experimenting and simulating probability models. Such investigations should embody a variety of realistic problems, from questions about sports events to whether it will rain on the day of the school picnic. Students should talk about their ideas and use the results of their experiments to model situations or predict events. Probability is rich in interesting problems that can fascinate students and provide settings for developing or applying such concepts as ratios, fractions, percents, and decimals.

Discussion

Probability, the measure of the likelihood of an event, can be determined theoretically or experimentally. Students in the middle grades must actively participate in experiments with probability so that they develop an understanding of the relationship between the numerical expression of a probability and the events that give rise to these numbers (e.g., 2/5 as it relates to the probability of choosing a red marble from a hat). Students must not only understand the relationship between the numerical expression and the probability of the events but realize that the measure of certainty or uncertainty varies as more data are collected.

Students have many misconceptions and poor intuitions about probabilistic situations. In order to bring these ill-formed notions to the conscious level so that they can be confronted, students should be asked to guess what will happen next or what the result of the experiment will show. An unexpected result has a much greater potential to cause students to rethink their basic assumptions if they have articulated their ideas before their experiment or analysis of the situation.

To see how the predictions we hear and see every day are based on probability, students must use their knowledge of probability to solve problems. In modeling problems, conducting simulations, and collecting, graphing, and studying data, students will come to understand how predictions can be based on data. Mathematically derived probabilities can be determined by building a table or tree diagram, creating an area model, making a list, or using simple counting procedures. Students develop an appreciation of the power of simulation and experimentation by comparing experimental results to the mathematically derived probabilities. For example, students can conduct experiments with two dice to determine the experimental probability of rolling a 4, construct a table to establish the theoretical probability, and then compare the two results. The following activity includes all these concepts and an analysis of fairness.

Arrange students in pairs and give each pair three chips: one chip with an "A" on one side and a "B" on the other; a second with "A" on one side and "C" on the other; and the third with "B" on one side and "C" on the other. One student tosses all three chips simultaneously onto the desk. Player 1 wins if any two chips match; Player 2 wins if all three chips are different.

After the students have decided whether to be Player 1 or Player 2 and have tossed the chips many times, they might want to revise their choices. When the class has completed the experiment and discussed the results, the theoretical probability can be analyzed by completing a tree diagram (fig. 11.1).

Fig. 11.1. Tree diagram

Player 1 can win in six ways, but Player 2 can win in only two ways; hence, the probability of Player 1 winning is 6/8 and the game is clearly unfair.

Students should also understand that some probability problems do not have theoretical solutions. Given a set of thumbtacks, what is the probability that one thumbtack will land "point up" when tossed? Many students will guess that the answer is 1/2. Through experimentation, they will discover that the probability changes with the size of the head of the tack and the length of the shank.

At the K-4 level, students can flip coins, use spinners, or roll number cubes to begin their study of probability. At the middle school level, such experiments should be extended so that students can determine the probabilities inherent in more complex situations using simple methods. For example, if you are making a batch of 6 cookies from a mix into which you randomly drop 10 chocolate chips, what is the probability that you will get a cookie with at least 3 chips? Students can simulate which cookies get chips by rolling a die 10 times. Each roll of the die determines which cookie gets a chip. The same type of simulation can help students determine how many boxes of cereal they should expect to purchase to receive at least 1 each of 6 types of prizes given away randomly, 1 to a box. With a computer or set of random numbers, this problem can be extended to simulate an industrial quality-control situation or to analyze the number of defective items that might occur under certain conditions on an assembly line. All these problems, experiments, and simulations can be easily studied, performed, and analyzed by middle school students.

Once students have experimented with a problem, a computer can generate hundreds or thousands of simulated results. It is important that the computer simulation follow active student exploration. This follow-up broadens students' understanding and provides them with an opportunity to observe how a greater number of trials can refine the probability model.

The nature of probability encourages a systematic and logical approach to problem solving. Throughout their experimentation and simulation, students should be making hypotheses, testing conjectures, and refining their theories on the basis of new information. Probability also can be applied to data analysis. Students can use charts, graphs, and plots to make predictions; this activity reinforces their interpretation of the information and their derivation of other useful information.

The table in figure 11.2 gives the record for Joan Dyer's last 100 times at bat during the softball season. She is now coming up to bat. Use the data to answer the following questions:

Fig. 11.2. Softball stats

What is the probability that Joan will get a home run?

What is the probability that she will get a hit?

How many times can she be expected to get a walk in her next 14 times at bat?

Probability connects many areas of mathematics. For example, fraction concepts play a critical role in the study of probability. Topics such as equivalent fractions, comparison of fractions, addition and multiplication of fractions, as well as whole number operations and the relationships among fractions, decimals, and percents can be reinforced through the study of probability.

Dividing the area of a rectangle into fractional parts to model a probability problem provides an excellent opportunity for students to identify the relationship between concepts in geometry and operations with fractions.

In the maze in figure 11.3, Tom is to pick a path at random. Use the grid in figure 11.4 to determine the probability that he will enter room A or room B.

Fig. 11.3. Maze

Fig. 11.4. Grid

To calculate the probabilities directly, students would have to multiply probabilities and then add the results. However, the use of an area model makes the problem more accessible. By using fractional parts of a region, a student can represent the sequence of possible choices for paths and then add simple fractions to determine the probabilities. P (A) = 1/6 + 1/6 = 2/6.