Day 20: Theoretical and Experimental Probability of Events: Part 3
Grade 8
Materials
• disks/tiles/cubes
(two colours)
• paper bags
• BLM 20.1
Assessment
Opportunities
Description
• Analyse a game of chance to demonstrate understanding of theoretical and
experimental probability.
Minds On…
Whole Class Æ Reflecting on Prior Learning and Orientating
students to Activity
Recall the concepts of theoretical and experimental probability discussed on
Day 19. Provide each pair with three green and three red tiles/cubes/disks (or
any two colours) and a paper bag. Introduce the game Green is a Go by having
students read BLM 20.1. Ensure that students understand how the game is
played.
Action!
Pairs Æ Investigation
Students play the game and each student completes all the questions. Working
with a partner, students consider changing the probability of the outcome. How
can the rules be changed in order to make the theoretical probability of winning
a 1 in 4 chance? (BLM 20.1, Answers).
Curriculum Expectations/Question and Answer/Mental Note: Listen to
pairs’ discussions, making mental notes of all of the ideas that need to be
discussed during whole class consolidation and debriefing.
Consolidate
Debrief
Whole Class Æ Demonstrate Understanding and Extend Thinking
Discuss the students’ answers to BLM 20.1 Green is a Go. As a class, decide
on the rules for a new game that will change the theoretical probability of
winning to 1 in 4.
Pairs Æ Game
In pairs, students conduct the new investigation. How does the experimental
probability compare with the theoretical probability the class discussed? (BLM
20.1, Answers.)
Encourage students to review the past few days’ work, in preparation for an
assessment (Day 22).
Concept Practice
Exploration
Reflection
Students may refer
to the results of
their coin toss
simulation.
Home Activity or Further Classroom Consolidation
Suppose you play the game with 3 green, 3 red, and 3 yellow tiles. Write a
summary in your math journal explaining how to find the theoretical
probability of drawing 2 green tiles from the bag. If you were to play the game
40 times, what result would you expect? Suggest possible reasons to support
your prediction.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 65
20.1: Green is a Go
Names:
Date:
With a partner, play a simple game involving six tiles in a bag, e.g., three red and three green
tiles. Take two tiles from the bag during your “turn.”
Rules
You may not look in the bag. Draw one tile from the bag and place it on the table. Draw a
second tile from the bag and place it on the table. Return the tiles to the bag.
You win if the two tiles drawn during your turn are both green.
Predict the number of wins if you play the game 40 times. Record and explain your prediction.
Play the Game
1. Take turns drawing two tiles from the bag, following the rules above. Record your wins and
losses on the tally chart. Continue this until you have played a total of 40 times.
Green, Green (win)
Red, Red (loss)
Red, Green (loss)
Totals
2. After you have played 40 times, use your results to find the experimental probability of
winning. (Remember that probability is the number of wins divided by the total number of
times the game was played.)
3. How does this compare with your predictions? Explain.
4. Find the theoretical probability of winning. (Hint: Use a tree diagram to show all possible
draws).
5. Write a paragraph to compare the theoretical probability you just calculated to the
experimental probability you found earlier. Are these results different or the same? Why do
you think they are the same/different?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 66
20.1: Green is a Go
(Answers)
Many students may predict 20 wins, thinking there are 2 possible outcomes, i.e., 2 green or not
2 green. Those with a bit more knowledge will predict 10 wins, basing their prediction on the
possibility of 2 heads resulting from the tossing of 2 coins. This would lead to thinking that the
possible outcomes are gg rr gr rg.
The table below shows all possible outcomes with tiles labelled g1, g2, g3, r1, r2, and r3.
1st pick / 2nd pick
g1
g2
g3
r1
r2
r3
n/a
win
win
loss
loss
loss
g1
win
n/a
win
loss
loss
loss
g2
win
win
n/a
loss
loss
loss
g3
loss
loss
loss
n/a
loss
loss
r1
loss
loss
loss
loss
n/a
loss
r2
loss
loss
loss
loss
loss
n/a
r3
Students may draw a tree diagram or list all possibilities. Students may show more or less
organization in their analysis of the outcomes, depending on their level of understanding.
There are 6 wins and 24 losses. Wins + losses = 30 (all possible outcomes).
6
1
Number of favourable outcomes
=
=
Probability =
Number of possible outcomes
30
5
Post-activity discussion:
Students may suggest different ideas to change the game to get a 1 in 4 chance of winning.
They may suggest rule changes or equipment changes. Each suggestion can lead to a rich
discussion or a new experiment to test whether it will produce the desired results and why it
does or does not. Some students may suggest placing only two of each colour in the bag.
1
However, a table of possibilities will show that this change leads to a probability of
for
6
winning. Based on this result and the emerging pattern of 2 of each colour yielding a probability
1
1
of winning of , and 3 of each colour yielding a probability of , students may suggest 4 of
6
5
3
.
each colour. However, this change of equipment yields a probability of
14
A suggestion that does not change the equipment for the game but strictly the rules (method of
play) is to pick the first tile from the bag, record its colour, return it to the bag and pick a second
tile. The analysis of this method of play is shown in the chart below. There are 9 wins out of the
36 total possible picks, producing the desired 1 in 4 chance of winning. From the discussion the
teacher can introduce the terms ‘with replacement’ (after the first tile is drawn out of the bag and
its colour noted, the tile is returned to the bag before the second tile is drawn from the bag) and
‘without replacement’ (one tile is drawn out of the bag and its colour noted; without returning the
drawn tile to the bag, a second tile is drawn from the bag and its colour noted).
1st pick / 2nd pick
g1
g2
g3
r1
r2
r3
TIPS: Section 3 – Grade 8
g1
win
win
win
loss
loss
loss
g2
win
win
win
loss
loss
loss
g3
win
win
win
loss
loss
loss
© Queen’s Printer for Ontario, 2003
r1
loss
loss
loss
loss
loss
loss
r2
loss
loss
loss
loss
loss
loss
r3
loss
loss
loss
loss
loss
loss
Page 67
20.2: A Probability Game
Name:
Date:
You want to develop a game, using red and green tiles, so that you have a 1 in 3 chance of
winning the game. Using up to 10 red and 10 green tiles, decide how many of each colour to put
in the bag and calculate the theoretical probability of drawing 2 green tiles.
Repeat this process by adjusting the number of red and green tiles until you arrive at a suitable
number of each colour in order to get the desired results of drawing two green tiles. Using the
number of each colour you decided on, play the game at least 30 times.
Use your data to compare the theoretical probability to the experimental probability. Explain why
there may be a difference between the two.
Green tiles
TIPS: Section 3 – Grade 8
Red tiles
Probability of drawing
two green tiles
© Queen’s Printer for Ontario, 2003
Decimal equivalents
Page 68
20.2: A Probability Game
(Answer)
As students work on their new game design, some may be content with probabilities that are
close to
1
; others may be exact. There are 121 possible combinations with 10 or fewer green
3
and 10 or fewer red tiles.
Access to a calculator or computer would be useful. If this is not possible you may need to guide
students to the conclusion to test only games with more green than red tiles.
Students may use charts, tree diagrams, or actual listing of combinations (as done in BLM
20.1). Be sure to allow adequate time for students to complete their work.
The combinations of red and green tiles that have probabilities close to or equal to
Green tiles
Red tiles
10
7
9
6
8
5
7
5
6
4
5
3
4
3
3
2
2
1
Probability of drawing
two green tiles
45
⎛ 10 9 ⎞
⎜ × ⎟ =
⎝ 17 16 ⎠ 136
8 ⎞ 12
⎛ 9
⎜ × ⎟ =
⎝ 15 14 ⎠ 35
14
39
7
22
1
3
5
14
2
7
3
10
1
3
1
:
3
Decimal equivalents
0.331
0.343
0.359
0.318
0.333
0.357
0.286
0.300
0.333
Students would likely only have time to find one or two combinations of red and green tiles to
lead to the desired results.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 69