D2. Probability
Describe the likelihood that events will happen, and use that information to make predictions.
Learning Situation: What Colour Will the Token Be?
Total duration: approximately 90 minutes
Summary
In this learning situation, students determine the experimental probability of randomly picking a red token and the probability of randomly picking a blue token from a bag containing a number of tokens. Students then play a game in which they use this probability to predict the colour of the token they will pick.
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Overall Expectation |
Specific Expectation |
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D2. Probability Describe the likelihood that events will happen, and use that information to make predictions. |
D2.2 Determine and compare the theoretical and experimental probabilities of an event happening. |
Learning Goals
The purpose of this learning situation is to have students:
- recognize that the results of an experiment may vary (variability)
- understand the relationship between the number of trials performed in a randomized experiment, the frequency of a particular outcome, and the probability of that outcome;
- explore the relationship between theoretical and experimental probability;
- apply the steps of problem solving to situations involving probability.
| Learning Context | Prerequisites |
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In previous grades, students have conducted simple probability experiments, compared the results to a list of expected outcomes, and described, in words, the probability of an event occurring. In Grade 5, students learn to express probability using a fraction and to compare the experimental probability to the theoretical probability. |
This learning situation allows students to explore the relationship between theoretical and experimental probability, and to discover that variability is inherent in any random situation. In order to complete this learning situation, students must be able to describe the theoretical probability of an event using fractions. |
Materials
- bag A: paper bag containing 3 blue tokens and 2 red tokens (1 bag per team of two)
- bag B: paper bag containing 5 red tokens and 5 blue tokens (1 bag per team of two)
- mystery bag: paper bags containing 35 red tokens and 15 blue tokens (1 bag per team of two)
- Appendix 6.1 (one copy per team)
- Appendix 6.2 (one copy per team)
- Appendix 6.3 (one copy per student)
Mathematical Vocabulary
probability, theoretical probability, experimental probability, frequency, possible outcomes, variability
Before Learning (Warm-up)
Duration: approximately 15 minutes
Prepare the A and B bags and the mystery bags in advance (see Materials). Introduce students to the following task:
"Today you will be conducting an experiment related to chance in order to explore important concepts in probability. To begin, you will do a small activity to review some of your knowledge of probability. I will give you a copy of the activity to do and two paper bags. I ask that you do not look at the contents of the paper bags and wait for me to give you instructions."
Group students into teams. Give each team an A bag, a B bag and a copy of Appendix 6.1 - Theoretical Probability and Experimental Probability.
Review the questions with the students. After reading a question aloud, ask a student to explain it in their own words. Ensure that everyone understands the task at hand. Review, if necessary, how to express a probability using a fraction. Then ask the students to answer the questions. Circulate in the classroom and intervene only for clarification purposes. When all teams have finished, review their answers and lead a discussion about the differences between theoretical probability (question 1) and experimental probability (question 3).
Example of Explanation
Question 1 - Bag A: The probability of randomly taking a token of a given colour from bag A is determined by looking at the contents of the bag. In fact, the tokens must be counted in order to establish a ratio between the number of tokens of a certain colour and the total number of tokens. So, by examining the contents of bag A, we see that the most likely outcome is to randomly pick a blue token and that the probability of randomly picking a blue token is equal to \(\frac{3}{5}\). Similarly, the probability of randomly picking a red token is equal to \(\frac{2}{5}\). When the probability is determined by viewing or counting the set of possible outcomes, it is called the theoretical probability.
Question 2 - Bag B: Since we can't look inside the bag, the only way to determine the probability of randomly taking a token of a given colour is to randomly take a token from the bag a number of times and record the results. The probability is then associated with the results of the experiment. This is an experimental probability. If, for example, a blue token was taken at random 7 times and a red token was taken at random 3 times, we say that the probability of taking a blue token at random is about \(\frac{7}{10}\) and the probability of taking a red token at random is about \(\frac{3}{10}\).
Active Learning (Exploration)
Duration: approximately 60 minutes
Present students with the following situation:
"I will give each team a different mystery bag that contains a number of red tokens and blue tokens. The game consists, first, of taking turns guessing the colour of the token you are going to pick at random. In the second stage, after you have picked the token at random, you check its colour. If you guessed correctly, you stay in the game; if not, you are eliminated. The last person to play wins the game. Before you play the game, I invite you to perform an experiment that might give you clues about the distribution of the colour of the tokens in the bag and help you win."
Group students into teams. Give each team a mystery bag containing 35 red and 15 blue tokens and a copy of Appendix 6.2 - Mystery Bag. Introduce students to the task at hand and tell them not to look inside the bag.
Allow students sufficient time to complete the experiment in Appendix 6.2. When all teams have finished, begin the game.
Suggested Game Play
- Ask all students to stand up.
- Choose a student at random to begin the game. Ask him or her to guess the colour of the token they are going to select at random and announce it to the class. Then ask them to take a token at random from their team's bag, show it to the class, and put it back in the bag.
- Write the colour of the randomly-selected token on the board.
- If the colour of the token does not match the announced colour, the student is eliminated from the game and must sit down.
- If the colour of the token matches the announced colour, the student remains standing and gets another turn.
- Move on to the next student until only one student is left standing. This student will be the winner.
Throughout the game, encourage students to analyze the results of the draws written on the board and to think about their game strategy. Pay attention to students' comments and reactions to assess whether they seem to understand and apply probability concepts. For example, after 10 consecutive draws of a red token, the student who is about to play decides to change his or her strategy because he or she thinks it is unlikely to randomly pick up a red token again. This reaction should be noted and a discussion about the concept of independent outcomes and the idea that "chance has no memory" could be facilitated later.
When the game is over, conduct a class review of the game by asking students questions such as:
- In your opinion, was the winner of the game lucky? Why?
- Would the same student win if we played again? Why?
Encourage students to justify their answer using mathematical arguments and express them using appropriate probability vocabulary. Encourage other students to respond to the arguments presented; for example, even if a red token was randomly selected 10 times in a row, there is no guarantee that a red token will not be randomly selected on the next draw.
Continue the discussion by asking students questions such as:
- What strategy did you use during the game? (I used our experimental probability data.)
- How many of you have used and relied on experimental probability to guess the colour of the token you were going to pick at random?
- Do you think that all teams had the same experimental probability? Why? (No, because the results obtained by each team depend on chance.)
Ask each team to reveal the results of their experiment (questions 1 and 2 in Appendix 6.2) and write the data in a relative-frequency table (including frequencies expressed as fractions).
Provide each student with a copy of Appendix 6.3 - What's in the Mystery Bags? Ask students to answer question 1 individually, taking into account the results written on the board.
Then invite a few students to give their answer, justifying their position. Encourage other students to respond to the arguments presented. Next, reveal to students that all the mystery bags contain 50 tokens and that they all have the same number of red and blue tokens. Point out that any probability experiment results in outcomes, some of which may be more likely than others, but that chance causes the outcomes to vary. Emphasize that variation in outcomes may be surprising, but it does exist, and that this phenomenon is called variability. Point out that, in data management, as in probabilities, variability must be taken into account in interpreting results. Have students individually answer question 2 in Appendix 6.3, starting with 50 tokens as the first sentence.
Consolidation of Learning
Duration: approximately 15 minutes
Ask a few students to present their estimate of the distribution of the tokens in the bag and to share with the class the reasoning they used to arrive at their estimate. Write the proposed estimates on the board.
The following are examples of possible estimations.
- When we did the 10 trials, we got 4 red tokens and 6 blue tokens. Since the bag contains 50 tokens, which is 5 times the number of trials, then the bag should also contain 5 times the number of red tokens and blue tokens. So I estimate that there are 20 red tokens (\(5\times 4\)) and 30 blue tokens (\(5\times 6\)).
- Since all bags have the same contents, the results of the game may represent results of draws from the same bag. During the game, we randomly drew mostly red tokens, so the bags probably contain more red tokens than blue tokens. So more than half of the tokens must be red. I deduce that there must be about 30 red tokens and 20 blue tokens.
- Since all the bags have identical contents, it is as if we had randomly taken a token from the same bag 120 times in the experiment (12 teams, each with 10 trials). According to the table, there were 86 red tokens drawn out of 120 draws, which is equivalent to 43 red tokens out of 60 draws \(\frac{86}{120}=\frac{43}{60}\). Since there are only 50 tokens in the bag, I would say that there are about 40 red tokens and 10 blue tokens.
Facilitate the discussion by asking questions such as:
- Do you think your estimate is a good approximation of the distribution of the tokens in the bag? Why?
- If I had not informed you that all the bags had the same contents, would you have made that same estimate? Why?
- Who do you think made the best estimate? What evidence makes you think it is more accurate than others? (I think that people who looked at more results to determine the number of tokens of each colour probably have a more accurate estimate, because looking at only the results of our 10 trials may be less representative of all the tokens in the bag.)
- Are you certain that your bag contains exactly this number of red and blue tokens? Why? (I cannot be certain, since I based my estimate on experimental probabilities.)
Then ask the students to:
- check the contents of their bag and compare it to their estimate of the distribution of the colour of the tokens (question 2 in Appendix 6.3);
- determine, using fractions, the theoretical probability of each result and compare them to the probabilities obtained during the 10 trials (question 2 in Appendix 6.2).
Facilitate a discussion about the conclusions that can be drawn from these two comparisons. Highlight, for example, the fact that, in several cases, there is a significant difference between the contents of the bag and the estimates of the distribution of the colour of the tokens in the bag. However, if we examine all the results obtained during the game or the 120 results obtained during the experiments (86 times a red token and 34 times a blue token), we see that, all things considered, the distribution according to the colours selected is quite similar to the actual distribution of colours in the bag. Have students recognize that the larger the sample size (for example, the set of game results), the more the distribution of the results obtained seems to be a good approximation of the distribution of the colours in the bag. Similarly, lead them to conclude that the larger the sample size, the more the experimental probability of any outcome seems to be a good approximation of the theoretical probability of the same outcome.
Differentiated Instruction
The activity can be modified to meet the needs of the students.
| To Facilitate the Task | To Enrich the Task |
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In the mystery bag activity, help students use the results of the experiment to determine the probability, using fractions, of drawing a red token and the probability of drawing a blue token. |
Have students simulate the drawing of the tokens using a graphing calculator or software. Prepare another mystery bag with some distribution of red and blue tokens. Have students collectively determine an experimental probability and then play the game again. Note: This time the game is different, because everyone has the same information at the beginning. |
Follow-Up at Home
At home, students can conduct an experiment with a family member to consolidate their learning about probability. They can do the following.
- Put 10 identical objects, such as tokens, marbles or cubes, in an opaque bag, three of which are one colour and seven of which are another colour.
- Have a family member randomly pick an object 10 times, noting its colour and returning it to the bag after each pick.
- Then ask them to estimate, among the 10 objects, the number of objects of one colour and the number of objects of the other colour.
- Examine the contents of the bag together, verify the estimate, and discuss it.
- Ask them to repeat the 10-object draw a few times and compare the results each time with that obtained the first time.
- Then explain the concepts of variability and probability.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 229-237.