NOTE: This is taken from the Teacher's Edition of What Do You Expect?. ©2002

What Do You Expect? mathematical and problem solving goals:

What Do You Expect? was created to help students:

The overall goal of the Connected Mathematics curriculum is to help students develop sound mathematical habits. Through their work in this and other probability units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: In what types of situations can probability be used to help make a decision? What are the possible outcomes for this situation? What techniques can be used to list all the possible outcomes?Are the outcomes equally likely? Can theoretical probabilities be calculated, or do I need to find experimental probabilities? How can I tell whether these events are independent or dependent? Is this game fair? If it is not fair how can the rules or the scoring system be changed to make it fair? If this game is played several times, what will be the expected value, or the average payoff?

The Mathematics in What Do You Expect?

Following is a summary of the basic ideas that are covered in the grade 6 probability unit, How Likely Is It?, and descriptions of the new mathematical ideas students will encounter in What Do You Expect?

Basic Probability Concepts

The term probability is applied to situations that have uncertain outcomes on individual trials but a predictable pattern of outcomes over many trials. For example, when we toss a fair coin, we are uncertain whether it will come up heads or tails; but we do know that, over the long run, we will get heads about half of the time and tails about half of the time. This does not mean that we can’t get several heads in a row, or that if we get heads on one toss we are more likely to get tails on the next. This concept—uncertainty on an individual outcome but predictable regularity in the long run is often difficult for students. Students often need a variety of experiences that challenge their prior conceptions before they grasp this basic concept of probability

If we toss a tack into the air, we know that it will land either on its head or its side. If we toss a tack many times, we can use the ratio of the number of times it lands on its side to the total number of tosses to estimate the likelihood that the tack will land on its side. Since this ratio is found by experimentation, it is called an experimental probability. Many uses of probability in daily life, such as weather forecasts and sports predictions, are based on experimental probabilities.

This unit offers many opportunities for students to collect data through experimentation and to use their data to assign experimental probabilities to the possible outcomes. It is important for students to realize that a small amount of data may contain a wide variation among the samples, and that only through experimentation over many trials can good estimates be made about what will happen in the long run. In other words, experimental probabilities must be based on a great number of trials relative to the number of possible outcomes.

In some situations, such as tossing a fair coin, we can also find theoretical probabilities. We know that a fair coin will land either heads up or tails up and that each outcome is equally likely. As there are two equally likely outcomes, the probability that a fair coin will land heads up is 1 out of 2, or ½. In general, the theoretical probability that a coin will land heads up can be expressed as follows:

Another example of a situation for which we can find a theoretical probability is the rolling of a six-sided number cube. The six possible outcomes — 1, 2, 3, 4,5, and 6 — are each equally likely to occur on any single roll. Thus, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6. We can use this theoretical probability to estimate that if a number cube is rolled many times, we could expect each number to be rolled about 1/6 of the time.

Probabilities, whether obtained through theoretical analysis or experimentation, are useful for predicting what should happen over the long run. Yet, a probability does not tell us exactly what will happen. For example, if we toss a coin 40 times, we may not get exactly 20 heads; but if we toss a coin 1000 times, the ratio of heads will be fairly close to ½. Experimental data gathered over many trials should produce probabilities that are close to the theoretical probabilities; this idea is sometimes called the Law of Large Numbers. If we can calculate a theoretical probability, we can use it to predict what will happen in the long run rather than having to rely on experi- mentation alone.

Theoretical Probability Models

Students who have studied the grade 6 probability unit How Likely Is It? have already learned quite a bit about conducting simulations to find experimental probabilities and making orga- nized lists of possible outcomes to find theoretical probabilities. In this unit, they will continue to work with these familiar strategies, while learning two new strategies for finding theoretical probabilities: making counting trees to list all possible outcomes, and constructing area models to represent the possible outcomes.

Counting Trees

Counting trees, introduced in Investigation 1 and used throughout the unit, offer students a way to determine all the possible outcomes in a situation systematically For example, suppose a spinner divided into three equal sections is spun and a six-sided number cube is rolled.

The possible outcomes can be shown in a list and a counting tree.

In this unit, students use counting trees to find the number of equally likely outcomes in situa- tions with a great number of possible outcomes. Counting trees are particularly useful for listing outcomes in situations involving a series of actions, such as rolling a number cube twice or rolling two number cubes; tossing a coin four times or tossing four coins; or choosing several items from a menu, such as a sandwich, a drink, and a dessert.

Counting trees can be used as a basis for understanding the multiplication of probabilities, though they are not intended to be used that way in this unit. Students do not yet understand enough about probability to know when and why it is appropriate to multiply probabilities. (Some of the ACE extension questions, however, are much easier to answer if students have discovered the idea of multiplying probabilities.)

Area Models

In How Likely Is It?, students divided the area of circular spinners to represent probabilities. Investigation 3 of What Do You Expect? lays the groundwork for thinking about probabilities in terms of area on a grid.

Area models, like counting trees, are useful for finding probabilities in situations involving successive actions, such as a basketball player who is allowed to attempt a second free throw only if the first succeeds. Unlike counting trees, an area model is particularly powerful in situations in which the possible outcomes are not equally likely. Whereas students will use counting trees to help determine the possible equally likely outcomes in a situation, area models will help them find and represent probabilities of outcomes that are not equally likely, as in multistage situations.

The area model below is a square divided to show the probability that a 60% free-throw shooter will score 0,1, or 2 points in a two-shot free-throw situation in basketball. In a two-shot situation, the player will get to attempt a second free throw whether or not the first free throw succeeds.

Expected Value

The long-term aspect of probability is a powerful concept. Rather than guarantee what will happen on a particular trial or even in the short run, probability models predict what will happen in the long run over many trials. Often, this is the most valuable information we can gain about a probability situation: a prediction of the expected value of the situation. The expected value is a long-term average of the probability of each outcome weighted by the payoff for that outcome.

In this unit, students are introduced to expected value in an informal yet concrete way We do not expect them to develop a formal definition of expected value or to use a formula for finding it. In fact, students might never use the term expected value in their work in this unit, instead thinking of the concept as “what to expect in the long run.” Expected value goes one step beyond basic probabilities because it incorporates the value—such as points earned in a game or money won in a contest—attached to each possible outcome in computing how many points or dollars we can expect to average per game or contest in the long run.

For example, consider the simple situation of a breeder of Labrador retrievers. Each male puppy will be sold for $200, and each female puppy will be sold for $250. In the long run, how much could the breeder expect to average per puppy?

We can approach the problem by analyzing a specific number of cases; let’s say 50 puppies born. Assuming that the probability of a male puppy is ½ and of a female puppy is ½, the breeder could expect about 25 puppies of each sex. This means the breeder could expect to collect 25 x $200 = $5000 for the male puppies and 25 x $250 = $6250 for the female puppies, for a total of $11,250. The average amount per puppy is thus $11,250 ÷ 50 = $225.

We could also arrive at this result by the computation ½($250) + ½($200) = $225, which shows each payoff weighted (multiplied) by the probability that it will occur. This strategy is closer to the mathematical definition of expected value but more conceptually difficult for students. (This idea is not directly addressed in this unit.)

Independent and Dependent Events

Toward the end of this unit, the idea of independent and dependent events is introduced informally This concept, a difficult one, is often a major focus of probability study in high school and college courses. Yet, we feel it is important to introduce this concept because many students working through a basic probability unit such as this one develop the belief that all events are independent.

Suppose you twice draw a marble from a bag containing two red marbles and two blue marbles. If you replace the selected marble after the first draw, the two draws will be independent of each other, because what you draw the first time will not affect what you draw the second time. If you do not replace the selected marble, the second draw will be dependent on the first draw, because the probability of drawing each color the second time depends on the color chosen on the first draw For example, if you draw a red marble the first time and do not replace it, the probability of drawing a red marble the second time is 1/3 rather than ½

In this unit, students analyze dependent events by using the situation to help make sense of the sequence of actions. They look at the context and determine the sequence of actions and the possibilities at each step in the sequence. The steps in the sequence guide the apportioning of the total area in an area model, or the designing of a counting tree representing all possible outcomes. Then, each portion of area in an area model, or each path on a counting tree, is compared to the total area or the total number of possible outcomes to form probability statements.

As students use an area model to make sense of two-stage probability situations, take any opportunity to help those who show readiness to see the connection to multiplying probabilities. As an area model is also used to develop an understanding of the multiplication of fractions, many students will see this connection naturally