Intent The gambler’s fallacy experiments lead into discussions of theoretical and observed probabilities and of independent events. In this activity, students begin thinking numerically about probability, using the scale for probabilities of 0 to 1 and distinguishing between theoretical and observed probability.

Mathematics The most important idea in this activity is that representing the probability of an event as number of outcomes you are interested intotal number of possible outcomes size 12{ { {"number of outcomes you are interested in"} over {"total number of possible outcomes"} } } {} is accurate only when all of these outcomes are equally likely. This definition of probability also helps students see why a probability must take on a value between 0 and 1, inclusive. Probabilities can be expressed as fractions, decimals, and percentages; students will be using all three forms in this unit.

Progression This activity begins with the teacher introducing the fraction definition of probability, focusing on equally likely outcomes.

Approximate Time 35 minutes for introducing and doing the activity 30 minutes for discussion

Classroom Organization Groups, followed by whole-class discussion

Materials Sentence strips

Doing the Activity

Part I: Finding Probabilities The focus in this activity is on building intuition. For some problems, a reasonable estimate should suffice, even if the problem has an exact theoretical answer. Encourage students to think in terms of both observed and theoretical probabilities. Begin by asking students, What are some examples of events that are impossible? Some examples of events that are certain? Keep a list of their suggestions. Then ask for examples of events whose chances of occurring are between “impossible” and “certain,” and list those as well. Next, draw a segment of a number line from 0 to 1.

Explain that mathematicians place “impossible” events at the left end of this number line and “certain” events at the right end. Then ask students to come up and indicate where they think some “between” events from the list should be placed. Introduce the “P(…) = ” notation for stating the probability of an event. For example, when a fair coin is flipped, it should come up tails half the time in the long run. This probability is written P(tails) = 12 size 12{ { {1} over {2} } } {} Point out that people often think of an activity or situation as potentially having one of several results. Introduce the term outcomes for describing these possible results. Ask, What are the possible outcomes for flipping a coin? (heads or tails) For rolling a die? (1, 2, 3, 4, 5, or 6) For handedness? (left-handed, right-handed, or ambidextrous) Ask the class how they could calculate probability when an event has only certain outcomes that can occur and they make the assumption that each outcome is equally likely. How can you calculate the probability of an event when the outcomes are equally likely? For example, one generally assumes that the two possible outcomes of a coin flip, and the six possible outcomes of a die roll, are equally likely. You might mention that probabilities based on such assumptions are examples of theoretical probabilities. Students should see that the probability of any one of these outcome is number of outcomes you are interested intotal number of possible outcomes size 12{ { {"1"} over {"total number of outcomes"} } } {} For example, for a die roll, there are six possible outcomes, each equally likely. So P(rolling5)=16 size 12{P \( "rolling" 5 \) = { {1} over {6} } } {}. We are often interested in a particular set of outcomes out of some larger set. For example, out of the six possible outcomes for a die roll, we might want to know the probability of rolling either 1 or 2. Elicit from the class the idea that, in general, the probability of getting a result that is within a specific set of outcomes you’re interested in is expressed by the fraction number of outcomes you are interested in total number of possible outcomes size 12{ { {"number of outcomes you are interested in"} over {"total number of possible outcomes"} } } {} You can illustrate with examples. For instance, ask, What is the probability of rolling 1 or 2 with a standard die? Students should be able to explain that there are six possible outcomes, and that since each is equally likely, P(rolling 1 or 2)=16 size 12{P \( "rolling" 5 \) = { {2} over {6} } } {} Acknowledge that it is not always clear whether the possible outcomes in a given situation are equally likely. Of the three examples given earlier—flipping a coin, rolling a die, and handedness—students will presumably know that the outcomes for handedness are not equally likely. Consider the question, What is the probability that a person is right-handed? Although there are three possible outcomes, the probability that a person is right-handed is much greater than 12 size 12{ { {1} over {3} } } {}. Students may assert that even the outcomes for coins and dice are not necessarily equally likely. Explain that, unless otherwise indicated, they should assume that coins and dice are fair—that is, all outcomes from a coin toss or a die roll are equally likely. Another example you might use is, What is the probability that someone was born in (your state)? Again, students should realize that the various possibilities are not equally likely. Students should recognize that any fraction of the type number of outcomes you are interested in total number of possible outcomes size 12{ { {"number of outcomes you are interested in"} over {" total number of possible outcomes"} } } {} must be between 0 and 1, because the numerator cannot be negative and cannot be larger than the denominator. Bring out the idea that if the set of “outcomes you’re interested in” contains all possible outcomes, you are certain to get one of the results you are interested in, and the fraction is thus equal to 1. Similarly, if there are no “outcomes you’re interested in,” the probability is 0. As you work with various examples, illustrate that probabilities can also be written as decimals and percentages. For example, students can write “P(tails)=.5 size 12{P \( ital "tails" \) ="50"%} {}” or P(tails)=50% size 12{P \( ital "tails" \) ="50"%} {}” instead of P(tails)=12 size 12{P \( "rolling" 5 \) = { {2} over {6} } } {} . Fractions, decimals, and percentages are all common ways to talk about probabilities, although the definition given above makes fractions the most natural form for the initial discussion.

Part II: Probabilities on the Number Line One nice way to set up this activity is to give groups identical strips of paper showing a number line marked from 0 to 1. Groups can write a large letter on their number lines to indicate the probability for each outcome. If the number lines are then displayed during the class discussion one directly below the other, students can see whether the letters line up. As you circulate among groups, you may see that a common misconception is that all outcomes in a situation are equally likely. This misconception often involves the way in which individual outcomes are identified. For instance, in question D, students may view the situation as involving three outcomes—flipping two heads, two tails, and one of each—and incorrectly conclude that each outcome has a probability of 13 size 12{ { {1} over {3} } } {}. You may wish to clarify this by returning to example A and arguing that because there are three colors, the probability of picking a blue gum ball is 13 size 12{ { {1} over {3} } } {}. Students will likely see that the colors are not equally likely to be chosen. If needed, help them recognize that it is more useful to think of each gum ball as a possible outcome rather than each color. Thus, they can think of the situation as having nine possible outcomes rather than three and see that the probability of getting a blue gum ball is 29 size 12{ { {2} over {9} } } {}. Returning to Question D, ask groups to find a way to express the problem in terms of equally likely outcomes.

Discussing and Debriefing the Activity You might begin by having groups display their number lines one above the other, so everyone can see how well the letters match up from group to group. Help students to distinguish between probabilities based on abstract models and probabilities based on observed or experimental results. For the latter, the probabilities are open to considerable interpretation. For example, you might ask, Are you basing your answer on observed data or on intuition? For example, has it ever snowed in Florida in July? Although there may be more disagreement on the problems involving observed probability, make sure students understand and can explain the mathematics in the simpler examples involving theoretical probability. Question A is an opportunity for the class to articulate again the definition of probability as number of outcomes you are interested in total number of possible outcomes size 12{ { {"number of outcomes you are interested in"} over {"total number of possible outcomes"} } } {} Emphasize that this formula applies only to situations with equally likely outcomes. Question B and C are based on observation, such as information gained from reading and weather reports. For Question D, it is unlikely all students will see why this example has a probability of 12 size 12{ { {1} over {2} } } {}. You might ask students to list the possible outcomes. The key is realizing that getting a tail and then a head is different from getting a head and then a tail. In other words, the equally likely outcomes are HH, HT, TH, and TT. Since the two flips are different in two of the four cases, the probability of getting different results is 24 size 12{ { {2} over {4} } } {}. For Question F, students may suggest that the teacher’s choices aren’t random. In this case, the problem concerns observed probability, and all sorts of answers are possible. If the two students are selected at random, the probability depends on the size of the class. For Question H, at least a few students will likely see from a theoretical model that the probability is 16 size 12{ { {1} over {6} } } {}. One approach is to imagine rolling one die first; in that case, the second die has a 16 size 12{ { {1} over {6} } } {} chance of matching the first. Another approach is to list all 36 two-die pairs (identifying them as equally likely), and note that 6 of these pairs are doubles. You may want to point out that the answer to Question H seems consistent with the result in Waiting for a Double, in which it takes an average of about six rolls to get a double. For Question I, students could use their experimental data from Waiting for a Double to estimate the probability. (Don’t expect students to be able to figure out the theoretical value, which is 91216 size 12{ { {"91"} over {"216"} } } {}, or about 42%.)

Key Questions What are some examples of events that are impossible? Events that are certain? Events between impossible and certain?

Supplemental Activities Mix and Match (reinforcement) asks students to determine the probability of choosing matching gloves from two drawers containing assortments of left-hand and right-hand gloves. The activity requires students to take into account that different outcomes may not be equally likely. Flipping Tables (extension) builds on the ideas in this activity and defines the terms combination and sequence.