### Purpose

To understand how probability is determined and why the number of times you try an experiment can affect the outcome.

### Context

In this lesson, by flipping coins and pulling marbles out of a bag, students begin to develop a basic understanding of probabilities, how they are determined, and how the outcome of an experiment can be affected by the number of times it is conducted.

Confusion about the application of probabilities in the real world can be alleviated by providing examples from medicine, natural catastrophes such as floods and earthquakes, weather patterns, sports events, stock market events, elections, gambling, and other topical contexts. (*Benchmarks for Science Literacy*, p. 226.)

At the K-2 level, students are likely to have had some experience in keeping track of different phenomena and in recognizing patterns in the outcomes of some. To this end, they are also likely to have had some experience in predicting the outcomes of certain events and providing reasons for them. By the time they reach the 3-5 grade level, they should also know that some things are more likely to happen than others and that some events can be predicted, while others cannot. In addition, they should know that they can often learn how to make predictions about groups' things by studying just a few of them. (*Benchmarks for Science Literacy*, p. 227.)

As with any scientific experimentation, students should become aware of the kinds of biases or methodology inconsistencies that can affect the accuracy of results. When students observe differences in the way things behave or get different results in repeated investigations, they should suspect that something differs from trial to trial and try to find out what. Sometimes the differences result from methods, sometimes from the way the world is. (*Benchmarks for Science Literacy*, p. 6.)

While teaching the lesson, it is important to be aware that some studies show that lower elementary children have no conception of probability, but others indicate that even lower elementary-school children have probabilistic intuitions upon which probability instruction can build. Another finding to be aware of is the misconception which all elementary-school children struggle with—the idea of representativeness—where an event is believed to be probable to the extent that it is typical. For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up. (*Benchmarks for Science Literacy*, p. 353.)

In general, although absolute certainty is often impossible to attain, we can often estimate the likelihood—whether large or small—that some things will happen and what the likely margin of error of the estimate will be. Students will benefit from also learning that it is useful to express likelihood as a numerical probability.

### Motivation

Begin the lesson by taking out a coin and showing it to the class. Then start flipping the coin, and ask:

- When have you seen people flip a coin?

Help them understand that people often do this to decide who goes first in a game or determine the outcome in competitions. Then ask:

- Why do you think people flip a coin to decide who goes first (when playing games)?

Guide student answers to the general conclusion that this is a fair way to make a choice. There are usually two people or teams in a game, and each team chooses a side of the coin—heads or tails. With the two sides of the coin equally likely to turn up, there is a 50 percent chance or probability that the outcome will be "heads" or "tails." Explain that the chance or probability can be expressed in other mathematical ways as well: 1 out of 2, 1 in 2, 1/2, or 1:2.

In a way that they will understand, also emphasize that while theory suggests that a flipped coin has a 50 percent chance of coming up heads, people who experiment with flipping coins and recording the outcomes usually will not get precisely 50 percent heads in an even number of flips. The more times one flips a coin, the more closely one approaches the theoretical 50 percent.

Then flip the coin a number of times, encouraging individual students to predict the outcome of the coin toss. Ask them, "Why did you choose heads (or tails)?" Reinforce the random outcome of the coin toss.

For greater exploration and reinforcement, you may decide to split the class into small groups and have them toss coins and try to predict the outcomes. Depending on the level of your students, you may also have them do a trial of 10 tosses, where beforehand they predict how many heads or tails they would get. Then have them perform the trials and keep a tally of heads versus tails. Discuss their findings.

### Development

To start, take out two of the three marbles (red and blue) and show them to the class. Place them into a dark bag (or shoe box), mix them up, and ask:

- With two marbles in this bag, what are the chances that I will pull out the red one (without looking)? Why?

Help students come to the same conclusions they arrived at with the coin: there is a 1 in 2 or 50 percent chance (or probability) of pulling one of the two marbles out of the bag. Once the similarities are clear, pull out the third marble (green), show it to the class, and place it in the covered box. Then ask:

- Now with three marbles in this bag, what are the chances that I will pull out the red one?

Help them to see that, in this case, since there are three similar marbles, the chance or the probability can be expressed as 1 out of 3, 1 in 3, 1/3, or 1:3.

**Marbles & Probability: 10 Trials**

Divide the class into small groups and hand out three different colored marbles and bags/boxes to each group. Also distribute the Marbles & Probability: 10 Trials student sheet. Explain to each group of "young scientists" that they will now test the 1 in 3 probability and will have to follow the procedures below carefully—since one slip up could offset the results! (Note: Before they start—for added fun and interest—each group might want to predict how many of each marble they will pull out.)

**Steps**

- Shake the closed bag with marbles inside.
- Pull out a single marble without looking.
- Look at and call out the color of the marble.
- Put the marble back in the bag.
- Record the results on the student sheet.

(Repeat Steps 1-5 until 10 marbles have been selected.)

For better accuracy and control, each group might be advised to assign these roles to group members: one who pulls the marble from the bag, one who calls out the color, one who records each result, and one who counts out each trial. Depending on the size of the groups, another person could serve as an "Experiment Leader" who overlooks and makes sure that these four critical functions run smoothly.

Each group should mark its sheet in the corresponding category for that marble. When finished—with teacher guidance—groups should tally their results (or probabilities) in the lower boxes of their 10 Trials student sheets.

** Sample Model:**

Red | Blue | Green | Total | |

Number of times marble was picked | 3 | 5 | 2 | 10 |

Results (#of picks divided by total # of picks) |
3/10 = 0.3 | 5/10 = 0.5 | 2/10 = 0.2 | 10/10 = 1 |

When finished, discuss and compare their findings, including these student sheet questions:

- Did your group follow each of the five steps carefully for each trial?
- Do you believe your results are accurate?
- How do your results compare with the theoretical probability of 1 in 3?
- Are you surprised by your results? Why or why not?

Record each group's results on the board (to be used for further analysis later). Also help students see that, in the case of this experiment, the picking of the marbles is purely random. However, in other experiments, other factors could contribute to inconsistent results (like the fact that one marble is bigger than another, that the marbles are visible, that data is incorrectly gathered, that an investigator is colorblind). All sorts of inconsistencies could bias the findings.

**Marbles & Probability: 50 Trials**

Next, pass out the Marbles & Probability: 50 Trials student sheet for the class to review. Explain that they will now test the 1 in 3 probability of their three marbles—but this time with a larger number of trials: 50. (Groups may again choose to predict the outcomes for each marble.)

Emphasize again the importance of accurate results, particularly with an experiment of this size. Have students follow steps 1-5 for each of the 50 trials and continue their specific roles in the experiment.

Once each group has completed 50 trials, have them tally and compute their results for each marble category. When finished, first discuss their individual group results (and record these findings on the board under their previous 10 Trials results).

- Did your group follow each of the five steps carefully for each trial?
- Do you believe your results are accurate?
- How do your results compare with the theoretical probability of 1 in 3?
- Are you surprised by your results? Why or why not?

Then discuss and compare each group's findings against other groups:

- How do your group results compare to ________ (other group)?
- Which group's results are closest to the probability of 1 in 3?
- Which group's results are farthest from the probability of 1 in 3?

Then compare their 10 Trials results with their 50 Trials results:

- Which is closer to the probability of 1 in 3, your 10 Trials or 50 Trials results?
- What do you think would happen if your group performed more trials? How would it affect the results?

### Assessment

As a way for students to review and further apply the ideas and concepts in this lesson, have them use the Marble Mania student esheet to visit the Marble Mania online interactive tool. This is a tool in which the randomness and probability of marbles being pulled out of a "bag" are calculated automatically and for rials up to 500. In short, students will be able to see how the experimental probability more accurately reflects the theoretical probability when larger numbers of trials are conducted (as compared to a few).

First, for simplification and better understanding, have students use the tool to repeat the lesson's first marble experiment scenario that was conducted with only two marbles in the box. (This can be done by putting "1"s in the red and blue boxes, and "0"s in the yellow and green boxes.) To make the experiment similar to the orginal, students should also select "1" marble to draw (per trial). Also make sure students click on the "Clear Trial" button after running each trial. Students can use the Marble Mania student sheet to record the results of their trials.

After students conduct this trial, discuss their results. Then allow them to increase the number of trials (50-100-500) under these conditions. They will notice that the frequency percentages (Frequency) get closer and closer to what the theoretical probability is.

Next, have students repeat their three-marble, 50-trial experiment using the apparatus and compare and discuss their results. Then have them perform the online experiment at 100 and then 500 trials. In each case, the frequency percentages should get closer and closer to the theoretical probability percentages.

Encourage a discussion of what all of this means. Questions may include:

- Which would most often come closest to the 1 in 3 theoretical probability—a marble experiment with 10 trials or 500? Why?

(As shown in their studies and noted in the first benchmark: "summary predictions are usually more accurate for large collections of events than for just a few.") - When conducting a scientific experiment like the marbles one, why is it important to conduct it carefully and use the same procedures?

(Answers may vary. In general, part of the reason is that results can become inaccurate if the same methodology or observation approach is not used in every trial.) - Why do you think scientists are interested in studying probabilities?

(Probabilities play an important role in medicine, predicting natural catastrophes such as floods and earthquakes, weather patterns, sports events, stock market events, elections, gambling, and other areas of everyday life.)

Finish the lesson by allowing students to experiment with the tool. One way to do this would be divide the class into pairs and assign each group one set of variables, for example, choosing four marbles and drawing two at the 10-50-100-500 trial levels. Students should be prepared to discuss with the class what their results suggest.

### Extensions

This lesson may be supplemented by other related Science NetLinks lessons involving uncertainty, probability, and scientific data collection and distributions:

Like the online marbles activity, PBS Math offers students a colorful online Virtual Coin Toss, where students can further test and reinforce what they know about the probable outcomes of a coin being tossed from 2 to 10,000 times.

The Math Forum offers access to numerous lesson plans, activities, online projects, and other resources involving probability and statistics for students of all levels.