Illuminations: Probability Explorations

# Probability Explorations

## What Are My Chances?

 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: Use probabilities to predict trends Interpret the relationship between experimental and theoretical probabilities Explore the Law of Large Numbers

### Materials

 Dice Pennies or other coins Several decks of Cards Calculators What Are My Chances? Activity Sheet

### Instructional Plan

In this lesson, students will conduct 5 experiments. If you have enough dice, coins, and decks of cards, students can work on this in groups. Otherwise, you can set up 5 stations, 1 for each game of chance, before class. Groups can rotate through the room, so that each group gets a chance to conduct each of the experiments.

Begin class by showing students a coin, and telling students they will be running experiments today and exploring probabilities. Begin the lesson with a discussion of theoretical probability. Write the formula for probability on the board:
 P(Event) = Number of Favorable OutcomesTotal Number of Outcomes
Discuss with students what the formula means. Then, ask what the chances of getting heads or tails are. When they say the chances are 1/2, write that down on the board. Next, show them a die. Ask what the chances of rolling a 4 are. When they say the chances are 1/6, write that down on the board. Continue this process for the other three games of chance, showing students a deck of cards, and asking the chances of picking a red card, a diamond, or the 5 of diamonds. If students do not know the probabilities, help them realize the correct solutions. Your board should now look something like this:
 Coin Die Card Color P(Heads) = 12 = 50% P(4) = 16 ≈ 17% P(Red) = 12 = 50% Card Suit Exact Card P(Diamonds) = 14 = 25% P(5 of Diamonds) = 152 ≈ 2%

Discuss with students that these are theoretical probabilities. Ask them to think-pair-share to come up with a definition for what theoretical probability is. Discuss the student explanations for theoretical probability and help them understand that theoretical probability is the likeliness of an event happening based on all the possible outcomes.

Next, ask students what they think the term experimental probability might mean. After working through the discussion of theoretical probability, students may be more willing to offer suggestions. Link back to the previous discussion by examining the term "theoretical probability." Theoretical probability has to do with the likelihood of events occurring in theory. It is what is expected to happen. Similarly, experimental probability has to do with calculating probability using the results of an experiment. The formula for probability can be rewritten to reflect this:
 P(Event) = Number of Favorable OutcomesTotal Number of Trials
In theoretical probability, you divide by the number of outcomes (e.g., There are 6 sides on a die so there are 6 possible outcomes). In experimental probability, you divide by the number of trials (e.g., If you only toss a die 4 times, then you divide by 4, not 6). Use the coin as an example. Toss it five times and record the number of heads. If it comes up heads 3 times, ask students what the experimental probability of getting heads should be. Students should agree it is 3/5. After this discussion, tell students they will be conducting several experiments to compare theoretical probability to experimental probability.

Distribute the What Are My Chances? activity sheet to students so they can follow along as you briefly explain what they will be doing for each experiment. Go through a single trial of each of the experiments and clarify any questions students may have.

Whether you break the class into groups working at their desks or set up stations, each individual students should conduct each of the 5 experiments. This will give more data to pool later in the lesson. As students conduct their experiments, you can tie in the theoretical probability by asking students questions like these as a reminder:
 What is the likelihood of getting a heads? [50%] What is the likelihood of getting a 4 on the die? [about 17%] What is the likelihood of picking a red card? [50%] What is the likelihood of picking a diamond? [25%] What is the likelihood of picking the 5 of diamonds? [about 2%]

When students finish the experiments in Questions 1–5, discuss their results and any observations they may have. Then, have students use the calculators to find the percentages for their experiments. Since the sample sizes are only 10 for each experiment, many will most likely not match the theoretical very well. This is expected and will enrich the discussion later when students combine all the class data. Ask students if it is useful or a good prediction for probability if they only use 10 samples. Have students share their thinking as to what number of trials may be needed to get a sample that could better be used to predict outcomes.

You can give several examples of times where small numbers are not good predictors of large numbers results:

• Would it be fair to give a report card grade based on 1 test? or 1 assignment?
• Would it be accurate to conclude that a coin will always come up heads after flipping it once?
• If 50% of students in a class said they like country music, do you think that means 50% of students in the whole school like country music?
• Could you assume that if a person throws a basketball once and makes a basket from half court, then they are a good shooter?

Combining their experimental results with the questions above, the class should agree that only a few trials is not enough to make predictions. This should motivate students to decide that combining the entire class data together will probably show more patterns. First, have students combine their group’s data , and then have the groups share their data on the board to combine everyone’s data together.

As a closure to this lesson, discuss and make comparisons with the students about the theoretical and experimental probability. Depending on your data, there should be a pattern as the experimental data begins to get closer to the theoretical calculations. It is possible that even with a class of data some results will still be far from the theoretical probability. If this arises, it should add to the discussion of the nature of probability. You never know what is going to happen with chance. Probability is just a tool to make predictions.

If time allows, use the Adjustable Spinner tool. As you look at it with your class, point out the experimental and theoretical probabilities as you spin multiple times. As the number of trials increases, the data percents will get closer to the theoretical probability.

### Questions for Students

 Is there a connection between theoretical and experimental probability? [Experimental probability will get closer to theoretical probability as more trials are conducted. This is called the Law of Large Numbers.] How could you explain the two types of probability to someone who has never heard of them? [Experimental probability is the chance of an outcome based on a performed event or experiment. Theoretical probability is based on what could happen theoretically if the event is to be performed.] Why is it useful to know about probabilities? [We can use them to understand events and what outcomes are possible, as well as what outcomes are likely.]

### Assessment Options

 Ask the students to write a reflection about theoretical and experimental probability are related and different. Have students design their own experiment and go through the questions on the What Are My Chances? activity sheet. Read to students a list of events such as the and have them decide if each is theoretical or experimental: Maria flipped a coin and got 6 heads out of 10 flips. Carlos said the chances of rain today are 30%. James said he has a 70% chance of making a free throw because yesterday he made 7 out of 10. 6 students out of 18 students in one classroom caught a cold, so the nurse said about 33% of the students in school would catch the same cold. Julia placed an eraser under one of four cups and told Patrick he had a 25% chance of finding the correct cup.

### Extensions

 Ask students to make a prediction table as to what they think the results will be through 30 trials of a coin toss. Have them number 1 through 30 on a piece of paper and record H or T for what they think will come up for each trial. Most students will end with experimental probabilities around 1/2 and will not put 6 or 7 heads in a row through the table, even though it is possible that the coin will land heads up for 7 in a row. Then, have students run 30-trial experiments and compare their predictions to actual trials. Discuss the word random and what it really means. Key question: If 7 heads come up in a row, does that mean tails is due? Introduce students to the Monte Carlo problem. Introduce the term equally likely events to students. Give them the example of tossing a die. The chances of rolling evens or odds are equally likely. Both have a 50% chance of happening. The probability of rolling a number less than 2, equal to 2, or greater than 2 are not equally likely. Rolling a number greater than 2 is much more likely. Read a list of events to students and have them respond whether each has equally likely outcomes. You could also have them place pennies on a scale between 0 and 100% where they think each of the probabilities will be. If the coins are in the same spot, the outcomes are equally likely. For example if you ask the chance of it raining today, students should place one coin to represent the chance of it raining and another to represent the chance of it not raining. If both coins are on 50%, then the events are equally likely. You may also need to discuss with students that the probabilities have to add up to 100%.

### Teacher Reflection

 Did the students understand the relationship between experimental and theoretical probability? Which method of finding probability do the students feel more comfortable with, experimental or theoretical? Why? Were concepts presented too abstractly? Too concretely? How would you change them? Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer? Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### NCTM Standards and Expectations

 Data Analysis & Probability 6-8Use 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.
 This lesson was prepared by Corey Heitschmidt as part of the Illuminations Summer Institute.

1 period

### Activities

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