MDM4U – Mathematics of Data Management (Grade 12)
LESSON: BASIC PROBABILITY CONCEPTS
Overall Expectations:
Solve problems involving the probability of an event or a combination of events for discrete sample spaces.
Specific Expectations:
1.4 determine, through investigation using class-generated data and technology-based simulation models (e.g.,
using a random-number generator on a spreadsheet or on a graphing calculator; using dynamic statistical software
to simulate repeated trials in an experiment), the tendency of experimental probability to approach theoretical
probability as the number of trials in an experiment increases (e.g., "If I simulate tossing two coins 1000 times
using technology, the experimental probability that I calculate for getting two tails on the two tosses is likely to be
closer to the theoretical probability of 1/4 than if I simulate tossing the coins only 10 times")
Learning Goal
Students will review empirical and theoretical probabilities and compare their results. They will see various ways to
implement a simulation to produce experimental results and understand what happens to empirical probabilities
as the number of trials increases.
1. MINDS ON
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Establishing a positive learning environment
Connecting to prior learning and/or experiences
Setting the context for learning
Whole class -> Discussion
What is Probability?
How is Probability Expressed?
What is Theoretical probability?
Find the probability of rolling an 8 on two dice.
Find the probability of rolling an even sum on two dice.
2. ACTION
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Introducing new learning or extending/reinforcing prior learning
Providing opportunities for practice and application of learning (guided -> independent)
Theoretical probability is calculated using mathematics based on the number of favourable outcomes divided by
the total number of possible outcomes. Of the 52 cards in a deck, only 4 are aces. The theoretical probability of
picking an ace is 4/52 or 1/13.
In experimental (empirical) probability, the probability is found by actually doing the experiment. For example,
determining how many times heads will come up if we flip a coin. We know theoretically that we have a 1 out of 2
chance of heads appearing (50%).
Small Groups -> Stations (DI)
Students work in groups at three centers:
Centre 1: At the smartboard (Flip, Spin, Roll) – Appendix A
Centre 2: Paper and Pencil (Rolling a die) – Appendix C
Centre 3: Using HP Mininotes (Flip, Spin, Roll) – Appendix A
Centre 4: Using HP Mininotes (Experimental Probabilities) – Appendix B
Students:
 Select the Learning Centre at which they feel they can learn best based on their Multiple Intelligences
preferences
 Summarize their observations discussion at the centre.
 Share their learning with the class
3. CONSOLIDATION AND CONNECTION
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Helping students demonstrate what they have learned
Providing opportunities for consolidation and reflection
You may be asking yourself, “How can the theoretical and experimental percentages be so different?” This is
because of a factor called randomness. Flipping coins is a random event. It’s entirely possible to flip the coin 10
times and have tails appear each time. Because of randomness no particular pattern as to how the coins will land
can be predicted. Theoretical probability tells us what we can expect to have happen based on the possible
outcomes. Experimental probability tells us what actually did happen. Theoretically, the more times you do the
experiment the more the experimental results should start to reflect the theoretical prediction.
Experimental (Empirical) Probability =
number of times the favourable outcome occurs
Total number of trials
Teacher -> SMARTBOARD presentation
Virtual Spinners and Coin Flippers (http://nlvm.usu.edu/en/nav/topic_t_5.html)
 Select Data Analysis & Probability (Grades 9 – 12)
 Select Coin Tossing.
Experiment using 10, 50, and 100 tosses. Are the results different? How do they compare to the
theoretical probability?
 Select Spinners
Spin 10 times. Spin another 10. Now spin an additional 100 time. Do the results get closer to the
theoretical probability the more times that you spin?
Individuals -> Exit Card
Students complete the Exit Card (Appendix C)
Collect reflections, provide feedback the following day and adjust instruction in response to Exit Card information.
Appendix A – Flip, Spin, Roll
OERB (Resource ID : ELO1205890)
Appendix B – Experimental probabilities
OERB (Resource ID : ELO1243370)
Appendix C- Paper and Pencil Experiment
Rolling a Die
1. Roll a die 10 times.
2. Record the results of each roll on a tally chart (frequency
distribution).
3. Graph the results using a bar chart.
4. Determine the experimental probability of rolling a 2 based
on your 10 rolls.
5. Write this probability as a fraction, a decimal, and a percent.
6. Roll the die another 40 times to make a total of 50 rolls.
7. Continue to record the results of each roll on the tally sheet.
8. Graph the results of the 50 rolls using a bar chart.
9. Determine the probability of rolling a 2 based on your 50 rolls.
10. Write this probability as a fraction, a decimal, and a percent.
11. How do the results of the 10 rolls and 50 rolls compare? Which one is closest to the
theoretical probability?
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Rolling a Die
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10
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0
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Appendix C – Exit Card
LESSON: BASIC Probability Concepts
Name _________________________________
1.
How many times do you expect to see heads appear in 60 tosses of a coin?
2.
What happens to the experimental probability as the number of trials increases?
3.
A student simulated the rolling of a die 1000 times using a random number generator on his computer.
After 50 rolls, a six occurred 20 times. Is there evidence that the random number generator has a fault?
Explain.