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 Outcomes
Total 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) = | 1
2 | = 50% | | P(4) = | 1
6 | ≈ 17% | | P(Red) = | 1
2 | = 50% |
| Card Suit | | Exact Card | |
| P(Diamonds) = | 1
4 | = 25% | | P(5 of Diamonds) = | 1
52 | ≈ 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 Outcomes
Total 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
1. 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.]
2. 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.]
3. 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.]
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?
Learning Objectives
Students will:
- Use probabilities to predict trends
- Interpret the relationship between experimental and theoretical probabilities
- Explore the Law of Large Numbers
Common Core State Standards – Mathematics
Grade 6, Ratio & Proportion
- 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.''
Grade 7, Stats & Probability
- 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.
Grade 7, Stats & Probability
- 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.