5. Propuesta de intervención didáctica
5.8 Previsión de evaluación
5.8.2 Indicadores de evaluación
Probability problems can often be represented in several ways. We have already seen how Venn diagrams provide a convenient way to visualise the members of various combinations of events.
Here, we introduce theprobability tree, also referred to as a tree diagram. This is best explained by way of an example.
Activity
The London Special Electronics company is investigating a fault in its manufacturing plant. The tests done so far prove that the fault is
at one of three locations: A, B or C. Taking all the test and other evidence into account, the company assesses the chances of the fault being at each site as:
Suspect site A B C
Probability of it being site of fault 0.5 0.2 0.3 The next test they will do is expected to improve the identification of the correct site, but (like most tests) it is not entirely accurate.
• If the fault is at A, then there is a 70 per cent chance of a correct identification; i.e. given that A is the site of the problem then the probability that the test says that it is A is 0.7, and in this case the probabilities of either possible error are equal.
• If the fault is at B, then there is a 60 per cent chance of a correct identification, and in this case too the probabilities of either possible error are equal.
• If the fault is at C, then there is an 80 per cent chance of a correct identification, and in this case too the probabilities of either possible error are equal.
Draw a probability tree for this problem, and use it to answer the following:
a) What is the probability that the new test will (rightly or wrongly) identify A as the site of the fault?
b) If the new test does identify A as the site of the fault, find:
i. the company’s revised probability that C is the site of the fault;
ii. the company’s revised probability that B is not the site of the fault.
Solution
Let A, B and C stand for the events: ‘fault is at A’, ‘fault is at B’ and
‘fault is at C’, respectively. Also, let a, b and c stand for the events:
‘the test says the fault is at A’, ‘the test says the fault is at B’ and ‘the test says the fault is at C’, respectively. The probability tree is shown overleaf in Figure 4.6.
a) The probability P (a) is the sum of the three values against
‘twigs’ which include the event a:
0.35+ 0.04 + 0.03 = 0.42.
b) i. The conditional probability P (C|a) is the value for the C ∩ a twig divided by P (a):
0.03
0.35+ 0.04 + 0.03 = 1
14 = 0.071.
ii. The conditional probability P (Bc|a) is the sum of the values for the A ∩ a and C ∩ a twigs divided by P (a):
0.35+ 0.03
0.42 = 0.905.
Alternatively,
P(Bc|a)= 1 −value ofB∩atwig
P(a) = 1 −0.040.42 = 0.905.
a 0.7
b 0.15
A c 0.15
0.5
a 0.2
B b 0.6
0.2
c 0.2
C a 0.1
0.3
b 0.1
c 0.8
p = 0.2 x 0.2 = 0.04 p = 0.3 x 0.1 = 0.03 p = 0.3 x 0.1 = 0.03 p = 0.3 x 0.8 = 0.24 Probability tree
p = 0.5 x 0.7 = 0.35 p = 0.5 x 0.15 = 0.075 p = 0.5 x 0.15 = 0.075 p = 0.2 x 0.2 = 0.04 p = 0.2 x 0.6 = 0.12 True site
of fault
Test says fault at
Final probabilities
Figure 4.6: Probability tree.
4.12 Summary
This chapter has introduced the idea of probability, and defined the key terms. You have also seen how Venn diagrams can be used to illustrate probability, and used the three axioms. This should prepare you for the following chapter.
4.13 Key terms and concepts
Bayes’ formula Conditional
Event Exhaustive
Experiment Independence
Mutually exclusive Partition
Probability Probability tree
Sample space Set
Tree diagram Venn diagram
4.14 Learning activities
A4.1 When throwing a die,
S = {1, 2, 3, 4, 5, 6}, E= {3, 4}, and F = {4, 5, 6}.
Give:
a) Fc; b) Ec ∩ Fc;
c) (E ∪ F)c; d) Ec ∩ F.
A4.2
Supplier Delivery time
Early On time Late Total
Jones 20 20 10 50
Smith 10 90 50 150
Robinson 0 10 90 100
Total 30 120 150 300
What are the probabilities associated with a delivery chosen at random for each of the following?
a) Being an early delivery.
b) Being a delivery from Smith.
c) Being both from Jones and late.
A4.3 Draw the appropriate Venn diagram to show each of the following in connection with learning activity A4.1:
a) E ∪ F= {3, 4, 5, 6};
b) E ∩ F= {4};
c) Ec= {1, 2, 5, 6}.
A4.4 There are three sites a company may move to: A, B and C. We are told that P (A) (the probability of a move to A) is 12, and P(B) = 13. What is P (C)? (Use the information given in the section underaxioms of probability.)
A4.5 Two events A and B are independent with probability 13 and 14 respectively. What is P (A ∩ B)?
A4.6 A company gets 60% of its supply from manufacturer A, the remainder from manufacturer Z. The quality of the parts delivered is given below:
Manufacturer % Good Parts % Bad Parts
A 97 3
Z 93 7
a) The probabilities of receiving good or bad parts can be represented by a probability tree. Show for example that the probability that a randomly chosen part comes from A and is bad is 0.018.
b) Also show that the sum of the probabilities of all outcomes is 1.
c) The way the tree is used depends on the information required. For example, show that the tree can be used to show that the probability of receiving a bad part is 0.028+ 0.018 = 0.046.
A4.7 (Using set theory and laws of probability or a probability tree.) A company has a security system comprising four electronic devices (A, B, C and D) which operate independently. Each device has a probability of 0.1 of failure. The four electronic devices are arranged so that the whole system operates if at least one of A or B functions and at least one of C or D functions.
Show that the probability that the whole system functions properly is 0.9801.
Solutions to these questions can be found on the VLE in the04a Statistics 1 area at http://my.londoninternational.ac.uk
In addition, attempt the ‘Test your understanding’ self-test quizzes available on the VLE.
4.15 Further exercises
Newbold, P., W.L. Carlson and B.M. Thorne Statistics for Business and Economics. (London: Prentice-Hall, 2009) seventh edition [ISBN 978-0135072486]. Relevant exercises from Sections 3.1–3.3 and 3.5.
4.16 A reminder of your learning outcomes
After completing this chapter, and having completed the essential reading and activities, you should be able to:
apply the ideas and notation involved in set theory for simple examples
recall the basic axioms of probability and apply them distinguish between the ideas of conditional probability and independence
draw and use appropriate Venn diagrams draw and use appropriate probability trees.
4.17 Sample examination questions
1. Say whether the following statement istrue or false and briefly give your reasons. ‘If two events are independent, they must be mutually exclusive.’
2. If X can take values of 1, 2 and 4 with P (X= 1) = 0.3, P(X = 2) = 0.5 and P (X = 4) = 0.2, what are:
a) P (X2<4);
b) P (X > 2 | X is an even number)?
3. Write down and illustrate the use in probability of:
a) the addition rule;
b) the multiplication rule.
4. A student can enter a course either as a beginner (73%) or as a transferring student (27%). It is found that 62% of beginners eventually graduate, and that 78% of transfers eventually graduate. Find:
a) the probability that a randomly chosen student is a beginner who will eventually graduate;
b) the probability that a randomly chosen student will eventually graduate;
c) the probability that a randomly chosen student is either a beginner or will eventually graduate, or both.
d) Are the events ‘Eventually graduates’ and ‘Enters as a transferring student’ statistically independent?
e) If a student eventually graduates, what is the probability that the student entered as a transferring student?
f) If two entering students are chosen at random, what is the probability that not only do they enter in the same way but that they also both graduate or both fail?
5. A coffee machine may be defective because it dispenses the wrong amount of coffee (C) and/or it dispenses the wrong amount of sugar (S). The probabilities of these defects are:
P(C)= 0.05, P(S)= 0.04, and P(C ∩ S)= 0.01.
What proportions of cups of coffee have:
a) at least one defect;
b) no defects?
Solutions to these questions can be found on the VLE in the04a Statistics 1 area at http://my.londoninternational.ac.uk
The Normal distribution and ideas of sampling
5.1 Aims
Now is the time where the ideas of measurement (introduced in Chapter 3) and probability (Chapter 4) come together to form the ideas of sampling. Your aims should be to:
understand the concept of a random variable and its distribution work with the Normal distribution
see the connection between the mathematical ideas of sampling introduced here and their uses (introduced in Chapter 2, and covered in detail in Chapter 9).
5.2 Learning outcomes
After completing this chapter, and having completed the essential reading and activities, you should be able to:
summarise the meaning of E[X] and Var(X)
compute areas under the curve for a Normal distribution state and apply the Central Limit Theorem
explain the relationship between size of sample and the standard error of the sample mean.
5.3 Essential reading
Newbold, P., W.L. Carlson and B.M. Thorne Statistics for Business and Economics. (London: Prentice-Hall, 2009) seventh edition [ISBN 978-0135072486] Sections 4.1, 4.3, 5.1–5.3, 6.1 and 6.2.
In addition there is essential ‘watching’ of this chapter’s
accompanying video tutorials accessible via the04a Statistics 1 area at http://my.londoninternational.ac.uk
5.4 Further reading
Wonnacott, T.H. and R.J. Wonnacott Introductory Statistics for Business and Economics. (Chichester: John Wiley & Sons, 1990) fourth edition [ISBN 978-0471615170] Chapters 4 and 6.
5.5 Introduction
If all governmental political and economic decisions were based on figures from entire populations, think how costly and
time-consuming that would be. You could argue that one would never in fact tackle the problem itself. If we needed to know exactly how many homeless families there are in a particular country before implementing policy, or the exact number of companies which employ people on a part-time basis, and for how many hours, then work might scarcely get off the ground! Think for example of the population of your country and the cost of a complete census, even if only limited questions are asked about residents.
Fortunately for everyone, statisticians have come up with a way of estimating such figures using samples from the population. They can also tell us how accurate these estimates are likely to be for a particular sample size and this is why survey sampling, rather than taking a full census, is the preferred tool of governments, social researchers and market research companies! The rest of this chapter develops the ideas you need in order to use random sampling successfully.