What's In The Bag?

Task 198 ... Years 4 - 8

Summary

One player secretly puts ten mixed coloured cubes in a bag. Without looking in the bag, the second player selects a sample of four cubes and records the colours. This sample is returned to the bag and the procedure of select and record is repeated twice more. The challenge is for the second player to work out what's in the bag.

Materials

40 objects in 4 colours
1 bag

Content

average
concept of proof
probability calculations
probability, conditional
probability experiences
probability, sample space / sample size
ratio & proportion
statistics, analysing data
statistics, collecting & organising data
statistics, confidence levels
statistics, inference

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The sampling techniques in this task are the same as in the 'real world'. Those who conduct opinion polls, or market research, or biological surveys usually can't check the whole population, so take samples and make inferences based on them. The mathematical decisions involved require application of probability theory and an understanding of concepts such as sample space, sample point, and event. Using the example on the card, here are some possible methods for predicting the population from the three samples.

	R	Y	G	B
Sample 1 ...	0	2	1	1
Sample 2 ...	0	3	1	0
Sample 3 ...	0	2	0	2

First Clues

There may be no red in the population.
There are at least Y = 3, G = 1, B = 2.
That's a total of 6 we can be sure of, so add 1 to each colour to make 10 in this ratio:
R = 1, Y = 4, G = 2, B = 3.

Add & Modify

Total the colours over the three samples: R = 0, Y = 7, G = 2, B = 3.
But there are only 10 cubes in the bag, so find ¹⁰/₁₂ of each number and round off to give: R = 0, Y = 6, G = 2, B = 2

Add, Order & Jiggle

Total the colours over the three samples: R = 0, Y = 7, G = 2, B = 3.
Put the totals in order: R, G, B, Y (0, 2, 3, 7).
Assign 1 to the smallest and jiggle the remaining numbers to make 10 and keep the order:
R = 1, G = 2, B = 3, Y = 4 is a reasonable guess, given there seem to be so many more Y than G and B combined. However, it could also be argued that there is no real evidence of red, so an adjusted prediction could be: R = 0, G = 2, B = 3, Y = 5

Probability & Proportion

Each sample is 4/10 of the population.
Multiply the elements of each sample by 2.5 (and round off) to find its proportional value in a population of 10. The results for each sample are then:
- Sample 1: R = 0, Y = 2, G = 1, B = 1
  ... Prediction 1: R = 0, Y = 5, G = 3, B = 3, which needs to be modified, because the total is greater than 10, to perhaps R = 0, Y = 5, G = 2, B = 3 when all three samples are considered.
- Sample 2: R = 0, Y = 3, G = 1, B = 0
  ... Prediction 2: R = 0, Y = 8, G = 3, B = 0, which needs to be modified, because we know there are blues, to perhaps R = 0, Y = 6, G = 2, B = 2 when all three samples are considered.
- Sample 3: R = 0, Y = 2, G = 0, B = 2
  ... Prediction 3: R = 0, Y = 5, G = 0, B = 5, which needs to be modified, because we know there are greens and blue seems too high, to perhaps R = 0, Y = 5, G = 2, B = 3 when all three samples are considered.

There are other possible methods. However, all methods are inexact. The case for one or the other as being most likely to be correct must be argued on the basis of evidence. The only way of being certain is to tip the bag out and count. Our students can do this, but a pollster researching opinions that represent the views of thousands of people can't 'tip the bag out and count' them all. They must also be prepared to present their research indicating their confidence levels. The calculation of confidence levels is further sophistication of statistical techniques.

Extensions

Suppose there was really one red block in the bag. What is the probability that it will not be selected in a sample of 4? If you had to give a percentage to indicate your confidence in the statement 'There are no reds in the bag.', what number would you assign?
Still using 10 in the bag from 4 colours, how would advise changing the sampling experiment to produce predictions more likely to be correct?

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
For this investigation you will need about 20 cubes in mixed colours and an appropriate container for each pair. You will also need a demonstration set and this works well if you use coloured plastic pegs and an ice-cream container. The pegs can be clipped to the edge of the container as they are withdrawn from the population.
Explain that you have 10 pegs in the container and that they came from a set you purchased that had four colours. Write up the colours. Challenge the students to guess the number of each colour. Guesses, will of course be wild. Ask the students how confident they are of their guess.
Would it help if I gave you a clue?
Tip out the container so that all are revealed.
Okay, that's your clue. Now make a guess. ... How confident are you of this guess?
Explain that finding out the colours of the pegs in the container is like work a mathematician does when finding out, for example, which of four political parties the people of Australia will vote for.
Do you think they could 'tip out the container' and ask every person in the country?
Suggest that today the investigation is about using information from a few samples to be able to predict, as accurately as possible, what is in the whole population.
Make up another secret set and ask students to help you withdraw four pegs without looking. Record the colours and have a brief initial discussion of the students' predictions. Replace the pegs and repeat twice more until there are three samples, as in the task.
Usually the number of samples a mathematician can take from a real population is governed by money or time. Let's suppose three samples of four is all we can do. Your challenge is to find a way to make a best guess at the population.
Allow time for groups to discuss, then list the range of responses. Give pairs time to rethink their strategy in the light of other people's and then each pair has to make a commitment to a prediction. Reveal what's in the container so students can check their prediction.
Provide materials for each pair to carry out their own experiments with a view to further testing and modifying their sampling strategy. When enough time has been provided for this initiate another class discussion to create an agreed set of say three successful techniques. Ask students to prepare an evidence-based report on the investigation which includes their proposal and the class set.
Explore what can be changed in the problem and ask students to record these questions that a mathematician might ask:

What happens if we change the population total but keep the same sample procedure, namely, three samples of 4 objects?

What happens if we change the number of selections in each sample, for example we use three samples of 5 objects?

What happens if we change the number of samples, for example we use four samples of 4 objects?

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 125, What's In The Bag?, which also includes two Investigation Guides - one of which explores confidence levels and one of which explores a point scoring system to test strategies.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The What's In The Bag? task is an integral part of:

MWA Chance & Measurement Years 3 & 4

MWA Chance & Measurement Years 9 & 10

The What's In The Bag? lesson is an integral part of:

MWA Chance & Measurement Years 9 & 10

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