Brief Description

The unit discusses the use of the breathalyser and mass radiography with the use of tree diagrams, the ideas of conditional probability and the occurrence of errors.

Design time: 4-5 hours

Aims and Objectives

On completion of this unit pupils should have some appreciation of random selection, Type I and Type II errors, conditional probability, and have been introduced to the relationship: P(A|B) x P(B) = P(A n B).

By the end of Section B they will have practised reading a two-way table, drawing tree diagrams and looking for multiplicative connections between probabilities. They will have used two-figure random number tables in a simulation, used tally marks to record results, and should be more aware of random variations.

In Sections C and D pupils are made aware that the same set of data can produce more than one tree diagram, whilst Section D gives practice in compiling a two-way table from a tree diagram in order to reinterpret the data.

Prerequisites

It is assumed that pupils have some familiarity with the idea of probability, especially as expected relative frequency, and that probability can be estimated from a sample relative frequency. Pupils will need to be able to express proportions as fractions and multiply fractions, simplifying where necessary.

Equipment and Planning

Two-figure random number tables are required in B2. Generally it is expected that pupils will work individually, but it may be helpful for them to work in pairs in B2, with one person reading the table, and the other identifying and recording the outcome. Class discussion of the italicized questions is to be encouraged. Shorthand notations for probabilities, e.g. Prob(A|B). Prob (A n B) have not been used in the pupil unit, but could be used where appropriate if desired.

An optional page R1 is provided to relieve pupils of the necessity of turning pages back and forth in order to find relevant tables and figures. This R page is not expendable, and sets could be retained for use on future occasions.

Detailed Notes

Section A

This section will require pupils to have an understanding of what the use of random tests would imply, and to discuss freely their opinion of such tests. If they record their opinions, these could be referred back to at the end of C4.

Section B

B1
This requires the pupils to read a two-way table, and to identify correctly the various cells of the table. The figures are not realistic; they were chosen for ease of handling.

The two types of error should become clear:

1. A person who is drunk does not react to the breathalyser and is regarded as sober no further charge will be brought on the grounds of drunkenness.
2. A person who is sober does react to the breathalyser, and will be required to accompany the policeman to the station to give a blood or urine sample, which is regarded as totally reliable.

Explanations as to how some of the sober people can react to the breathalyser depend on how recently they have had a drink. Interesting comments on this can be found in the AA Book of the Road.

B2
If there is pressure on time, this simulation could be omitted, but the variation that can occur in random testing is interesting. If it is thought necessary to have some means of checking the pupils' ability to use a table of random numbers correctly, then it would be possible to direct them all, individually or in groups, to start at the same position in the table, and use them in the same way. If it is thought necessary to aggregate the results to obtain a sample larger than 100, care must be taken to see that the table is used differently by different groups or individuals, to avoid compounding occasional abnormalities.

Where the tables have been used differently by individuals or groups, it would be interesting to compare answers to c and d across the class to bring out the variation that even random selection can produce.

B3
The point should be stressed that it is the proportion of errors that is important.

The start is made here on the ideas behind conditional probability, and pupils need to be clear in their own minds at this stage, before going any further, of the distinction between the proportion of drunk drivers who gave a positive reaction, and the proportion of all drivers who were both drunk and gave a positive reaction, etc.

Questions a to e use the results in Table 1, so the answers can be checked easily. There may be a case for answering questions a to e as a class exercise. Question f is reinforcement; pupils use their own results.

B4
Tree diagrams, as used here, need to show the proportions as well as the absolute numbers. Multiplication of proportions along the branches should produce the figures in the 'Proportion of total' column, and is one way of checking the accuracy with which pupils have recorded their own figures.

B5
It may be necessary to stress the fact that, from a small sample of only 100 trials, it is only possible to estimate the probabilities. For a more able group, it would be interesting to compare the proportions obtained, on which the probabilities are estimated, with the original probabilities used with the random number tables.

B6
While Probability (a motorist is drunk) and Probability (a drunken motorist gives a positive reaction) are given, it is necessary at this stage to look back at Table 2 to get Probability (a motorist is both drunk and gives a positive reaction), unless the multiplicative property has been fully appreciated by this time.

Section C

C1
Here the more likely outcome of using the breathalyser is investigated. This section encourages pupils to read the original table in another way, and leads on to a different form of tree diagram, where the original analysis is on the basis of positive or negative reaction to the breathalyser. It should be made clear to pupils that the entries in the column 'Proportion of total' in Figure 2 are the same as those in Figure I, but their order of occurrence is necessarily different.

Pupils should be made aware that if the breathalyser alone is used, then those drivers who give a negative reaction will be regarded as sober, and the twigs at the end of the second branch will never be drawn.

4. 'The most important error' depends on who is making the judgment. A drunk driver may be glad to get a negative reaction, but other road users may not be so happy. A sober driver who gets a positive reaction is subjected to extra worry and time wasting.
5. Improve the breathalyser; but this may mean it is less easy and more expensive to use.

C3
This could eventually be extended to include a formal introduction to Bayes' theorem in the form:

using the probabilities obtained in B6.

C4
There is an opportunity here for the effects of other probabilities on the extent of the errors to be considered. The use of 1000 people should limit the occurrence of fractions.

C5
This is intended to be an open-ended section and one that will depend for its success and effectiveness on the appreciation by the pupils of the likely outcomes from other ways of selecting which drivers to test. It will be necessary to consider P(drunk|another offence has been committed) and P(another offence has been committed|drunk). Theoretical values will have to be assigned to these probabilities before any useful discussion can take place. With a less able group this may not be feasible. It is necessary to point out to all pupils the inherent diAiculties in taking a truly random sample of all motorists. Even systematic sampling of every tenth vehicle, with the first vehicle selected by means of random selection, limits the pop- ulation to those motorists who happen to be on that road at that time, and may, if carried out strictly in accordance with instructions, lead to motorists who commit a traffic offence not being breathalysed.

Comparison of individual answers to C4c and their first answer in A could prove interesting.

Section D

D1
This section requires pupils to work backwards from a tree diagram to a two-way table. It also contains a social message: that positive reaction to mass radiography does not necessarily imply the existence of tuberculosis (TB). The suggestion of considering 100000 people is to make it more likely that the proportions suggested by the probabilities will be applicable, and to ensure that no fractions occur in the table. Less able pupils may need a calculator to help with the arithmetic.

7. A person with TB does not react (has clear photograph). A person who doesn't have TB does react (has mark on the photograph). The first is the more serious error. Mention the effect on the person, contact with other people, etc.

Answers

B1

1. 29
2. 31
3. Six regarded as sober
4. Eight regarded as drunk - need to take further test

B3

1. ⁸/₁₀₀
2. ⁶/₁₀₀
3. 6, ⁶/₂₉
4. 71, 8, ⁸/₇₁
5. ⁶³/₇₁

B4

5. (²⁹/₁₀₀) x (⁶/₂₉) = (⁶/₁₀₀); (⁷¹/₁₀₀) x (⁸/₇₁) = (⁸/₁₀₀); (⁷¹/₁₀₀) x (⁶³/₁₀₀) = (⁶³/₁₀₀)

B5

1. ⁸/₇₁
2. ⁸/₁₀₀
3. ⁷¹/₁₀₀
4. ²³/₁₀₀
5. ²³/₂₉
6. ²⁹/₁₀₀
7. (⁸/₇₁) x (⁷¹/₁₀₀) = (⁸/₁₀₀) (a x c = d)
   P(+ve|sober) x P(sober) = P(+ve and sober)
   (²³/₂₉) x (²⁹/₁₀₀) = (²³/₁₀₀) (e x f = d)
   P(+ve|drunk) x P(drunk) = P(+ve and drunk)

B6

1. P(drunk) x P(+ve|drunk) = P(+ve and drunk)
2. ⁷⁰/₁₀₀
3. ¹⁰/₁₀₀
4. ⁷/₁₀₀
5. (⁷⁰/₁₀) x (¹⁰/₁₀₀) = (⁷/₁₀₀)
   P(sober) x P(+ ve|sober) = P(+ ve and sober)

C1

1. 31
2. 23
3. 6
4. ⁸¹/₃₁
5. ²³/₃₁
6. ⁶/₆₉
7. ⁶³/₆₉
8. ²³/₁₀₀
9. ⁸/₁₀₀
10. ³¹/₁₀₀
11. (⁸/₃₁) x (³¹/₁₀₀) = (⁸/₁₀₀) (d x j = i)
    (²³/₃₁) x (³¹/₁₀₀) = (²³/₁₀₀) (e x j = k)

*C3

B5g gives P(+ve|sober) x P(sober) = P(+ve and sober)
C1k gives P(sober|+ve) x P(positive) = P(+ve and sober)
giving P(+ve|sober) x P(sober) = P(sober|+ve) x P(positive)

D1

5. 89.911, 1
6. 10 089, 9990
8. P(has TB|+ve reaction) = (⁹⁹/₁₀₀₉₈) = 0.0098 (2 sig. figs)

Page R1

Table 1 - Results of 100 tests of the breathalyser.

Figure 1 - Tree diagram showing proportions obtained from Table 1.

Figure 2 - Tree diagram showing further results from Table 1.

Test Questions

100 people were given a blood test, and 40 of them were found to be drunk. A breathalyser was then tested by these 100 people. 909o of the drunk people reacted positively, while 80% of the sober people reacted negatively. These results are to be summarized in a tree diagram.

Figure 1 - Tree diagram showing outcome of 100 tests.

1. Copy Figure 1. Fill in the ends of the branches and complete the column headed 'Probability'.
2. a Find the Probability (a driver is both drunk and reacts positively).
   b Find the Probability (a driver is drunk).
   
   The Probability (a drunk driver reacts positively) is ³⁶/₄₀.
   c Write an equation involving ³⁶/₄₀ multiplication, and your answers to a and b.
   d Write the same equation, using probabilities instead of fractions.
3. Use the same tree diagram to find:
   a Probability (a driver is sober),
   b Probability (a driver is both sober and reacts positively to the test),
   c Probability (a sober driver reacts positively to the test).
   d Write an equation involving multiplication and your answers to a, b and c.
   e Write the same equation, using probabilities instead of fractions.
4. Copy Table 1 and use the information from your tree diagram to complete it.
   

   	Reacts positively	Reacts negatively	Total
   Drunk	36		40
   Sober
   Total			100

   Table 1 - Outcome of 100 tests.

5. Copy Figure 2 and use the information in Table 1 to complete it.
   
   Figure 2 - Testing the breathalyser.
6. A driver was stopped and asked to take a breathalyser test.
   a What two types of error could occur?
   b According to Figure 1, which was the most likely to occur?
   Suppose 100 motorists were stopped.
   c How much of Figure 2 could be drawn?

Answers

1. 
   Figure 1.
2.  
   1. Probability (a driver is both drunk and reacts positively) = ³⁶/₁₀₀
   2. Probability (a driver is drunk) = ⁴⁰/₁₀₀
   3. (³⁶/₄₀) x (⁴⁰/₁₀₀) = (³⁶/₁₀₀)
   4. Probability ( a drunk driver reacts positively) x Probability (a driver is drunk) = Probability (a driver is both drunk and reacts positively) i.e. P(D|+ ve) x P(D) = P(D n + ve)
3.  
   1. Probability (a driver is sober) = ⁶⁰/₁₀₀
   2. Probability (a driver is both sober and reacts positively to the test) = ¹²/₁₀₀
   3. Probability (a sober driver reacts positively to the test) = ¹²/₆₀
   4. (⁶⁰/₁₂) x (⁶⁰/₁₀₀) = (¹²/₁₀₀)
   5. Probability (a sober driver reacts positively to the test) x Probability (a driver is sober) = Probability (a driver is both sober and reacts positively to the test) i.e. P(S|+ve) x P(S) = P (S n +ve)

4. 	Reacts positively	Reacts negatively	Total
   Drunk	36	4	40
   Sober	12	48	60
   Total	48	52	100

5. 
   
   Figure 2
6.  
   1. A drunk driver can have a negative reaction: he will be allowed to go free. A sober driver can have a positive reaction: he will need to have a blood test.
   2. Probability (a drunk driver reacts negatively) = ⁴/₄₀ = ¹/₁₀
      Probability (a sober driver reacts positively) = ¹²/₆₀ = ²/₁₀
      The second type of error is more likely to occur.
   3. All but the last two twigs at the end of the second branch.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 4)

Choice or Chance
Retail Price Index
Figuring the Future
Sampling the Census
Smoking and Health
Equal Pay

Units at Other Levels In the Same or Allied Areas of the Curriculum

Level 1

Shaking a Six
Being Fair To Ernie
Probability Games
If at first...

Level 2

On the Ball
Fair Play

This unit is particularly relevant to: Social Science, Mathematics, Sciences.

Intereonnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

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