This unit looks at, analyses and extends a game of chance. Two red and two blue multi-link cubes are placed in a bag. The player takes out two cubes. If the player gets two red cubes they win.
- find a theoretical probability
- use more than one way to find a theoretical probability
- check theoretical probabilities by trials
- identify what a fair game is and how to make an unfair game fair
In this unit we are essentially looking at five ideas. These are how to calculate theoretical probabilities, how to test probabilities by trialling, how to generalise to arbitrary numbers of objects, that theoretical probability and experimental estimates of probability may not be equal and how to make games fair.
We show that probabilities can be calculated directly from the definition by determining appropriate outcomes, by using the product of two probabilities, and by tree diagrams. We encourage students to use more than one method here as it deepens their understanding and gives them more than one strategy to use in new situations. Sometimes it is not possible to provide a theoretical probability so it is important to be able to determine an estimate of probability by using trials (experimental estimates of probability). Generalisation is an important skill in all mathematics. So we encourage it here to extend students' experience of finding patterns. The more able students may be able to justify the generalisation. Students should know when games that they are playing are fair as they may otherwise lose a lot of money in later life. So we experiment here with some simple games and see more than one way to make a game fair. All of the above skills will be useful in later statistical and mathematical work.
For a similar unit to this see Heads and Tails, Level 4.
Sufficient multi-link cubes so that each group of students can have 5 red, 5 blue and 5 green cubes
Paper bags to draw the cubes from
experimental estimates of probability, theoretical probability, trialling, generalise, estimate, tree diagrams, possibilities, chance, outcomes, combinations, fairness
Here the students are to draw two multi-link cubes from a bag in which there are two red and two blue cubes. A student wins if they take two red cubes.
Now Jo says that there are three outcomes: red, red; blue, blue; and red, blue. So the chances of winning are one in three.
On the other hand, Brian says that the chance of getting a red cube is 1 in 2. So the chance of getting two red cubes is ½ by ½ = ¼.
(Incidentally this will be more interesting if you use names of people that are known to the students in your class.)
The unit is based on the following 10 questions.
Question 1: Is Jo right? Why?
Question 2: Is Brian right? Why?
Question 3: Are they both wrong? If so, what is the probability of getting two red cubes?
Question 4: What argument can you give to convince the class that one of the answers above is right? Are there any other arguments that you can use to give the same result?
Question 5: What experiment can you do to support the verbal argument?
Question 6: If the above questions are too hard, can you see a simpler situation that will give you some insight into what is happening with the first problem?
Question 7: If you are convinced that you understand the problem, consider this game. Jo and Brian play against each other. If Jo draws two red cubes she wins. When you put the cubes back in the bag, Brian draws two cubes. If Brian gets a cube of each colour, he wins. Is this a fair game? (In other words are they both equally likely to win?)
Question 8: Repeat the above with three red cubes and two blue cubes and draw two cubes. What is the probability of getting two red cubes; or one red and one blue; or two blue cubes?
Question 9: Suppose that there are 120 red cubes and 110 blues cubes. What are the chances of getting two red cubes when you draw two cubes?
Question 10: What if there are r red cubes and b blue cubes? What is the probability of getting two red cubes; a red and a blue cube; or two blue cubes?
Now you may not get as far as Question 10 but there should be enough material here to stimulate students through 5 sessions. We now look at the answers to all of the questions and then look at a possible teaching sequence.
Of course you can go even further and have three different colours of cubes. (Or four, or five, …)
Question 1: Jo is wrong. The situation is more complicated than she thinks but she has the bones of an idea that will work. If she can nail the correct outcomes she’ll be in business. We’ll see how to do this as we go further.
Question 2: Brian is wrong. He has overlooked the fact that when you remove a cube from the bag, the probability of getting the next cube has changed.
Question 3: Yes they are both wrong. The probability should be 1/6.
Question 4: Investigate Jo’s argument a little more deeply. What are the possible outcomes? The problem is that the cubes act as individuals even though some of them have the same colour. If we think of the red cubes as being R1 and R2 and the blues as being B1 and B2, then the outcomes are R1R2, R1B1, R1B2, R2B1, R2B2, B1B2. (Note that R2R1 is the same outcome as R1R2.) So this says that there are 6 equally likely outcomes. So the chance of getting two reds is 1/6.
On the other hand Brian wasn’t so far off. Certainly the chance of getting a red is 1/2 (just check the possible outcomes). But when you take out one red, there are three cubes left. Only one of these is red. So the chance of getting the second red is 1/3. Then the chance of getting two reds overall is 1/2 x 1/3 = 1/6.
You can also do this using a tree diagram if you know how to construct one. The tree diagram can be based on either of the arguments above. So you might have 12 endpoints if you use the first idea and 6 if you use the second.
Question 5: Just do it! Get students to draw two cubes from a bag and record what happens. If each pair does this 30 times the data can be grouped to give a whole class data set. This won’t show that two reds appeared one sixth of the time but it should be closer to 1/6 than to 1/3 or 1/4.
Note that this gives evidence. It doesn’t give a proof!
Question 6: Try with one blue cube and two red cubes. Analyse that to see what happens.
Question 7: Using any of the arguments of Question 4, we see that the probability of getting a red and a blue cube is 2/3. The game is certainly not fair and Brian will win much more often than Jo.
Question 8: Two reds: 3/5 x 2/4 = 6/20= 3/10; red and a blue: 3/5 x 2/4 (red first) + 2/5 x 3/4 (blue first) = 12/20 = 6/10; two blues: 2/5 x 1/4 = 2/20= 1/10.
Note that here Jo and Brian have a fair game (see Question 7).
Question 9: Two reds: 120/230 x 119/229; red and a blue: 120/230 x 110/229 (red first) + 110/230 x 120/229 (blue first); two blues: 110/230 x 109/229. YUK!!
Question 10: Two reds: r/(r + b) x (r – 1)/(r + b – 1) = r(r – 1)/(r + b)(r + b – 1); red and a blue: r/(r + b) x b/(r + b – 1) (red first) + b/(r + b) x r/(r + b – 1) (blue first) = 2br/(r + b)(r + b – 1); two blues: b/(r + b) x (b – 1)/(r + b – 1) = b(b – 1)/(r + b)(r + b – 1).
You might like to note here that this means that r(r – 1) [number of ways two reds can be selected] + 2br [number of ways a red and a blue can be selected] + b(b – 1) [number of ways two blues can be selected] = (r + b)(r + b – 1) [number of ways two cubes can be selected]. So the sum of the different combinations has to equal the total number of combinations.
In this lesson we look at the cubes’ problem.
- Introduce the basic situation of the bag and the four cubes.
- Tell the class about Jo and Brian.
- Ask them who they think is right and who is wrong. Take a class vote. There are three possibilities here: Jo right; Brian right; neither right. Check that everyone has voted.
- Now go through Jo’s argument.
How does it sound?
Is it OK?
If it’s not, what’s wrong with it?
- Now go through Brian’s argument.
How does it sound?
Is it OK?
If it’s not, what’s wrong with it?
- Does anyone think that neither Brian nor Jo have correct arguments?
What do you think the probability is of getting two red cubes?
(Don’t ask for an explanation at this stage.)
- Give the students time to work in pairs to discuss the problem.
What is the probability of two red cubes? Why?
Ask them to write down their arguments.
- Bring the class back together. Repeat the vote of step 3 above.
- Let different students say what they think the probability is. Get them to argue their cases.
- Repeat the vote of step 3.
In this lesson, we go further with the problem by experimenting to see what might be the exact probability. Then we consider four ways to get the right answer.
- Recall the problem from the last session. We want to know what the probability of getting two red cubes is from a bag containing two red cubes and two blue cubes.
- What do you think the probability of getting two red cubes is?
- Just collect possible answers and write them so that all the students can see the possibilities.
- How can we get evidence to support one probability from all the possibilities?
Make sure that they realise that if they take an experimental approach, the answer they get is unlikely to be exactly the same as theoretical probability.
- Let the students work in pairs to collect evidence by drawing two cubes out of a bag. They should do about 30 trials; record the number of successful trials; and divide the successful trials by the number of trials. Any pair that finishes more quickly than the rest of the group can do more. Walk round to check that everyone has understood the task correctly.
- Bring the students together and list their results separately in three columns (successful trials, number of trials, quotient of these two) on the board. Combine the results to give a class data set. Divide the number of successful trials by the number of trials.
Which suggested theoretical probability is this number closest too?
Why isn’t it exactly the same as any of the theoretical probability?
Is there any difference between the different groups’ data and the combined data?
Why would you expect that?
- Now return to the arguments for the different theoretical probabilities. Discuss the four arguments (see Question 4 above with the two different tree diagram arguments) for the correct probability.
- Which of these arguments do you find most easy to understand? What aspects of the others do you find hard to see?
- Let the students write down their favourite argument in their books. Check this at some stage.
In this session, we build on the arguments of the last two sessions and extend finding probabilities to other situations.
- Recall the problem of the two red cubes and get the students to recall the arguments that were used to find the probability of finding two red cubes.
- Send the students into their groups to find the probability that you draw two blue cubes and that you draw one red and one blue cube. Ask the students to find at least two arguments for each of these probabilities.
- Bring them back to discuss their results.
- Get them to work in pairs again with three red cubes and two blue ones. Ask them (i) what the possible outcomes are when two cubes are drawn and (ii) to provide two arguments to justify their arguments.
- Have a reporting back session to make sure that everyone is on track.
- This time send the pairs off to work together with different numbers of red and blue cubes. Ask them to do the same two things that they did in step 4.
- Have a reporting back session. Put their results systematically (perhaps using a table) on the board.
- Ask them what the probabilities would be for some ridiculously big numbers (such as the ones in Question 9).
Can you justify these results?
Can you generalise to r red cubes and b blue ones? Discuss.
Today we look at the game played by Jo and Brian in Question 7.
- Recall what has been happening regarding drawing cubes from bags.
- Tell them about the game of Question 7.
What do you think about this game?
Do Jo and Brian have an equal chance of winning?
So this is not a fair game.
- From now on, we play the game and Jo wins if she gets two red cubes while Brian wins if he draws a red and a blue cube. Let the groups go away to consider who wins if there are (i) 3 red cubes and 2 blue cubes; (ii) 5 red cubes and 2 blue cubes; and (iii) 7 red cubes and 2 blue cubes.
- Bring the groups back to discuss the results that they found. Let them argue their cases. Resolve any disputes by making the students provide convincing arguments in their defence. Encourage the students to use a range of arguments. Only if necessary, step in to correct specious arguments.
- Is it possible to find a rule so that we can tell straight away from the number of red and blue cubes who will win the game?
What is the relation between the number of red and blue cubes for the game to be fair? How could you prove this?
- At this point you may feel that it is worth checking what you have discovered with particular numbers of red and blue cubes. Alternatively you might like to introduce green cubes into the game. (This doesn’t change the relative probabilities of getting two reds and getting a red and a blue.)
Today we find other ways to make the game fair.
- Go back to the original game in Question 7. Is there a way to adjust the rules so that the game becomes fair? Let students discuss this in their groups.
- As a class, discuss the ideas that they produce. One solution might be to give Jo a number of lollies every time she draws two reds and to give Brian a different number of lollies.
What would these numbers be?
- Another solution would be to give Jo more turns than Brian.
How many turns should Jo have compared to Brian?
- Can you think of any other solutions?
- Now play the game in groups with varying numbers of red and blue cubes. Based on the numbers of red and blue cubes they choose, get each group to say what rules they are using to make the game fair before they start playing. Play the game 30 times to see what happens.
- Get the groups to report back on the numbers of cubes they used; what they thought would make the game fair; and how the games panned out in the 30 trials.
Did each person come out equally?
Was that because your game wasn’t fair?
Why else could it be?