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.

Figure 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) = 1212 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 outcomesnumber 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)=16P(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 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)=16P(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 1212 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

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)=.5P(tails)=.5 size 12{P \( ital "tails" \) ="50"%} {}” or P(tails)=50%P(tails)=50% size 12{P \( ital "tails" \) ="50"%} {}” instead of P(tails)=12P(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 1313 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 1313 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 2929 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.

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.