This unit explores a variety of activities (games of chance) where the number of outcomes in the sample space is clearly different from the number of events. Students learn to see the difference between events (for example a score of 4) and outcomes (for example 3 and 1, 2 and 2, 1 and 3).
The concept of chance may be viewed as at variance with the causal, logical and deterministic thinking that characterises much of mathematics. Through life experiences students develop their own views of probability. These views are often not well developed and they may interpret chance to mean that nothing mathematical may be stated about a situation. Therefore, learning about probability poses particular difficulties, meaning, students at all levels harbour misconceptions about probability.
Many students interpret probability as synonymous with the proportion of times they think an event will occur. In turn, they may expect the event to always occur in that proportion. Compound events present particular difficulties, especially when they require a distinction to be made between “and” and “or” in a probability context. Very few students have any intuitive approach to finding the probability of a compound event, so situations need to be explored fairly formally. For many students, traditional approaches of defining the sample space and listing the number of favourable outcomes do not result in understanding. Even constructivist approaches, based on dice games where the scores from two dice are added, often result in misconceptions. Many students can see that there are more ways of getting a score of seven than of getting a score of two, but still cannot calculate the probabilities.
This unit explores the concept of sample space, using activities where the difference between the number of events and the number of outcomes is clearer than in many games used in the past.
It is a good idea to give students a format for recording their work. Basing an investigation on the statistical PPDAC cycle is useful, using the headings Problem, Plan, Data (from the experiment), Analysis (including theoretical probability), and Conclusion. Alternatively, students can develop a theoretical model based on their understanding of the situation, and follow that by testing the model with an experiment. In that case, PPDAC can still be used, with Plan including the development of the theoretical model. Students will need to record experimental results systematically – tally charts are the easiest method. Requiring students to record their initial hunch as part of the initial Problem stage can engage them in the idea of carrying out an investigation and having a stake in the conclusion.
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to support students include:
The context for this unit can be adapted to suit the interests and experiences of your students. For example, your students might be motivated to explore the fairness of games that are played in their cultures or households, or that were historically popular in different cultures.
Te reo Māori kupu such as putanga (outcome), hoahoa rākau (tree diagram), putanga whakamātau (experimental outcome), tūponotanga whakamātau (experimental probability), putanga tātai (theoretical outcome), tūponotanga tātai (theoretical probability), and tūhuratanga tauanga (statistical investigation) could be introduced in this unit and used throughout other mathematical learning.
Grid method
Explore the grid method with the following prompts:
Red | Red | Red | Blue | |
Red | RR | RR | RR | BR |
Red | RR | RR | RR | BR |
Blue | RB | RB | RB | BB |
Blue | RB | RB | RB | BB |
Tree diagram method
Explore the grid method with the following prompts:
Probability tree approach
Probability trees are focused on at Level 7 of the curriculum. It does not need to be introduced at Level 5. Even if some students have suggested it, it might be best to treat it as extension material either later or for some interested students. Students need to see the equivalence of a probability tree with the tree diagrams of method 2. Teaching them to multiply along branches without understanding is counter-productive.
Whichever theoretical method or methods has been used to determine the probability of snap, this probability should now be compared with the long-run frequency that has been found experimentally. In probability it is desirable to use theoretical and experimental approaches to verify each other.
Is the answer the same from both approaches?
Why do you think they are different?
How close does the experimental answer have to be before we accept that it is confirming our theoretical answer?
How do the probabilities compare with your original hunch?
There is always a finite chance that the long-run frequency might be quite different to the expected answer. All that we can do in this situation is to collect more data to obtain a longer run.
Is the game fair?
How do we know it is fair?
What is the probability of winning?
How do we know that this is the probability?
Have students work in pairs or small groups to explore a number of variations on the game. Are any of the following games fair?:
Start with a class discussion of how we can determine whether each of the games is fair. Ensure that students are clear that they need to determine whether there are equal chances of winning and losing. Discuss also the need to use both theoretical and experimental approaches in order to verify solutions.
If students use an experimental approach, scaffold them with questions like:
If students use the grid method to investigate the theoretical probability, scaffold them with questions like:
If students use the tree diagram method to investigate the theoretical probability, scaffold them with questions like:
Hold a class discussion where students present the results from their investigation into a variation of marble snap.
This is a game for two players.
The same approach used with “Marble Snap Revisited” may be used here also. Start with a class discussion of how we can investigate the probabilities of the game. Ensure that students are clear that to know whether a game is fair they need to determine if there are equal chances of each player winning. Also discuss the need to use both theoretical and experimental approaches in order to verify solutions. The same format should be used for recording their work, such as the PPDAC cycle. Discuss the desirability to record the number of times each score is rolled, rather than just the number of times a player wins the game.
If students use an experimental approach, scaffold them with questions like:
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequencies for 0, 1, 2, 3, 4 and 5.
If students use the grid method to investigate the theoretical probability, scaffold them with questions like:
If students use the tree diagram method to investigate the theoretical probability, scaffold them with questions like:
It is useful to record how often each event occurs rather than just which player won, as it enables the number of times each event occurs to be compared directly with the theoretical probability for that event.
There are many different ways in which the game can be altered to make it fair.
P(0) = 6/36
P(1) = 10/36
P(2) = 8/36
P(3) = 6/36
P(4) = 4/36
P(5) = 2/36
We might give player A the odd numbers, or just 1 and 2, or 0, 1 and 6 etc.
The AND/OR game is a game for two players, A and B, played with a grid like the one below, and three counters. The game requires two spinners like the ones below, with paper clips to spin on them, or two modified dice with sides 1,1,1,1,1,3 and 1,1,1,1,1,5. Students will follow the PPDAC cycle to investigate the OR game first, determining whether the game is fair. Then they will investigate the AND game, and compare the probabilities of the two games.
What problem could you investigate about the probability of the OR game? What problem could you investigate about the probability of the AND game?
The OR game: The spinners are both spun (each player can spin one spinner, using a pen to hold a paper clip in the centre of the spinner, the paper clip is then flicked, recording the sector it lands in), or the two dice rolled. The product of the two resulting numbers is found. If the product is 1 OR 15, then player A wins. If the product is 3 OR 5, then player B wins.
What is your hunch about who is most likely to win?
The AND game: The spinners are both spun or the two dice rolled. The product of the two resulting numbers is found, but in this game each player must get both their numbers to win. Player A must get 1 AND 15 to win. Player B must get 3 AND 5 to win. If the product is 1 or 15, then player A covers that number with a counter. If the product is 3 or 5, then player B covers that number with a counter. The first player to cover both of their numbers wins.
What is your hunch about who is most likely to win?
If students use an experimental approach, scaffold them with questions like the following:
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequency.
If students use the grid method to investigate the theoretical probability, scaffold them with questions like:
If students use the tree diagram method to investigate the theoretical probability, scaffold them with questions like:
The OR game is a straightforward game like Dice Differences, where the event of a player winning is determined by one event. The AND game is a game with complex probabilities. The chance of winning the AND game is primarily determined by the probability of the rarest of the possible events, since common events like (1, 1) are repeated without affecting the probability of winning. Working out the probability of winning the AND game requires a more sophisticated use of probability trees than is available to students at level 5 in the curriculum. Using a probability tree which ignores repeated outcomes (since they don’t affect who wins) the probability of player A winning is 0.2396 (4dp) and 0.7604 (4dp) for player B winning, but students are not expected to work this out. The key concept here is for students to understand that the probabilities of AND games are different to the probabilities of OR games, and to realise that in some situations, experimental estimates of probability are the best way to find the probability of an event.
A final activity is “the three coin game”. This is also a game for two players and is played on a board as illustrated below. It could also be used as an assessment.
One person to be A, and the other is B.
Play the game on this board, using a counter and a coin:
Rules of the game:
Students could investigate “the three coin game” using the PPDAC cycle, recording their hunch about the probability of a Player winning.
What problem could you investigate about The three coin game?
What is your hunch about the probability of player A winning?
If students use an experimental approach, scaffold them with questions like:
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequencies for wins for Player A and Player B. Students should realise that the game is not fair, and that some locations on the game line are impossible.
If students use the grid method to investigate the theoretical probability, scaffold them with questions like:
If students are using the tree diagram method to investigate the theoretical probability, scaffold them with questions like:
Hold a class discussion where students present the results of their investigations into The Three Coin Game.
Dear families and whānau,
As part of our mathematics programme we have been investigating the results of theoretical and experimental approaches to games of chance. Ask your child to teach you one of the games we have explored in class, and to tell you about the probability of winning the game
Printed from https://nzmaths.co.nz/resource/fair-games at 11:33am on the 25th April 2024