## Probably Graphing

• Lesson
• 1
• 2
6-8
1

Student will conduct a coin tossing experiment for 30 trials.  Their results will be graphed and shows a line graph that progresses toward the theoretical probability.  The graph will also allow for a representation of heads or tails throughout the experiment.

Begin this activity by asking all students to stand up. Tell them you are going to flip a coin. If they think it is going to be heads, they are to put a hands on their head. If they think the coin is going to be tails, they are to put their hands behind their back. If they are correct, they remain standing, if they are incorrect, they sit down. Repeat the process until there is only one person left. If you choose to do so, then that person can receive a prize.

During this introduction, discuss any patterns in the results of the coin toss and the number of students who sit down each time. Relate this to the previous lesson (What are my Chances?) by asking students what the theoretical probability is of getting the correct answer. [50%.]

Also, keep count of how many heads and tails were tossed during the game. Ask students to find the experimental probability and compare it to the theoretical. Ask students:

• What chance did you have of being correct on the very first flip of the coin?
[ or 50%.]
• What chance do you have of being correct the second time?
[50%. You have to choose between heads and tails each time. Each time your chances will be , regardless of previous flips.]
• Does a previous flip have an effect on upcoming events?
[No. Each flip is independent of previous flips. You may wish to introduce the term independent events.]

Inform students that this lesson will continue to investigate the probabilities involved in a coin toss. Distribute the Probably Graphing Activity Sheet and a penny to each student. Read through the introduction together. Students may need help in understanding that if they get a tails, they carry over the previous number in the Heads row.

As you read through activity sheet and before students begin the experiment, draw the coordinate plane on the board. Ask students to discuss what they feel the graph will look like. Have a few students come up and draw their best guesses on the board. Some students may draw a graph similar to the ones below. Don’t confirm or deny any responses. Instead ask students if they agree or disagree with the predictions.

Inform students that when they create their own graphs, they should plots the points representing their data, then connect the points with line segments. Explain that the lines are there to help determine the trend, but do not represent possible points on the graph like a graphed line normally does. It is not possible to flip a coin 1.5 times.

When all questions have been addressed, allow students time to conduct their experiments and graph their results. Once completed, ask students to compare and contrast their results with students in their group or around them. Find out if there is anyone who had an extremely large number of heads, or low numbers of heads occur. Have students hold up stand up and walk around holding their graphs in front of them and comparing to others as they walk by. Most of the graphs will probably resemble the pattern below.

If some students have graphs that do not approach 50%, discuss their graphs with the class. Does this mean their graphs are wrong? [No.] Remind students of the discussion in the previous lesson (What are My Chances?) about how the number of trials affect the probability.

From the most of the graphs, it should be apparent that with low trial numbers the graph fluctuates greatly. One trial may move the line 20–30%, while with nearly all trials completed the graph may only move 1–4%. This again re-emphasizes that small numbers of trials will not be a good predictor of the theoretical probability. As the number of trials increases we begin to see a graphical representation of the Law of Large Numbers. The experimental probability will approach the theoretical probability of the event. In this case, it approached a probability of 50%.

As this experiment progresses it would take several heads in a row to create a noticeable spike. This can lead to a discussion of whether several heads in a row is likely, even though randomly possible. You can look at the data from your students and see what student had the most repetitive results of heads. For example, one student may have had 6, 7 or 8 in a row. Even if they didn’t, you can discuss how results like this would affect the graphs. [They would create a spike, until a tail occurs.] Even if there was an instance of 7 in a row, the graph will still begin to draw closer to the theoretical probability of 50% with more added trials. This can be a fun discussion because you can never actually say it’s impossible; it is just highly unlikely or not probable!

### X/O Problem

For a class of high achieve students, you may wish to conclude with the following situation. It is best if you show them actual cards and perform the experiment a couple times to allow them to see it performed. Ask them to debate the probability of the situation. This should create some great discourse.

You have three cards marked as follows:
• One card with an X on both sides
• One card with an O on both sides
• One card with an X on one side and an O on the other

Suppose all three cards are in a bag. You reach into the bag randomly draw a card, and you are looking at an X.

• Is it more likely that the other side will show an O? an X? Or are both equally likely?
• What do you think is the theoretical probability of an X being on the other side?

Most students will not know how to find the theoretical probability. It is a problem that creates great debate, even among professionals. However, by running an experiment of many trials, the Law of Large Numbers should give a reasonable approximation of the theoretical probability. In this way, the problem allows for discussions and connections between experimental and theoretical probability, as well as showing the power of the Law of Large Numbers.

The correct probability is . If you wish to explain to students, present this representation where each X is numbered. The 3 possible situations are therefore:

1. If you see X1, then X2 is on the back. (This represents the X/X card.)
2. If you see X2, then X1 is on the back. (This represents the same X/X card, but flipped over.)
3. If you see X3, then an O is on the back.(This represents the X/O card.)

Of the 3 situations, 2 result in an X on the reverse side of the card. Therefore, the probability of having an X on the reverse side is . Many people will think it is actually a 50/50 chance because you have 2 cards with X's, the X/O and X/X cards, and 1 card without X's, the O/O. Since you already know the O/O is eliminated, they think it is a 50/50 chance that an X is on the back. But the difficulty is you don’t know which X you are looking at to start with, so there are three possibilities as explained above.

Assessment Options

1. Ask students to sketch what a graph of what the experimental probability graph would look like for spinning a 1 on a spinner we three equally-sized sections labeled 1, 2, and 3.
[The graph would most likely spike up and down and eventually even out around 33%.]
2. Ask students to write a reflection defining theoretical and experimental probabilities and how they are similar and different.
3. Have students design their own experiment, perform a number of trials, and graph their results.

Extensions

1. Show students how to use and create tree diagrams for modeling possible outcomes. For example, flipping a coin twice (shown in the diagram on the left) or flipping a coin and rolling a die (shown in the diagram on the right).

1. Ask students to perform and graph the X/O problem for 30 trials to compare the experimental and theoretical probabilities.

Questions for Students

1. Why does experimental probability approach theoretical probability?

[As more trials are conducted at random, unexpected outcomes such as having several heads in a row will eventually even out. Also, after several trials, each successive has less and less influence on the experimental probability. For example, if after 100 trials there have been 43 heads, the experimental probability is 43%. Then, after the 101st flip the experimental probability will be either ≈ 43.6% if heads comes up or ≈ 42.6% if tails comes up.]

2. Give a definition of the word random.

[Having no specific pattern or influence on an experiment's outcome.]

3. Can a formula be used to simulate a random event? For example, can a computer be truly random?

[Formulas can create outcomes that appear random. However, they are never truly random since they are by definition created by a systematic process. Computers, therefore, cannot be random. They can only appear random.]

Teacher Reflection

• What was your students’ level of enthusiasm/involvement? How could you increase it?
• How did you challenge the high-level achievers?
• How did the students demonstrate understanding of probability?
• Which concepts presented too abstractly? too concretely? How would you modify the instruction so that the transition from concrete to abstract is more fluid?
• How did you set clear expectations so that students knew what was expected of them? What would you do different next time?
• What adjustments, if any, did you find necessary to make adjustments while teaching the lesson?

### What Are My Chances?

6-8
Students will conduct five experiments through stations to compare theoretical and experimental probability. The class data will be combined to compare with previously established theoretical probability.

### Learning Objectives

Students will:

• Graph probabilities to investigate trends.
• Interpret the relationship between experimental and theoretical probabilities.
• Explore the Law of Large Numbers.

### NCTM Standards and Expectations

• Understand and use appropriate terminology to describe complementary and mutually exclusive events.
• Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.
• Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

• CCSS.Math.Content.7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

• CCSS.Math.Content.7.SP.C.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.