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SADC Course in Statistics Introduction to Probability and Demography Ideas (Session 01)

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To put your footer here go to View > Header and Footer 2 Module Objectives At the end of this module, you will be able to explain basic concepts of probability theory describe several commonly occurring probability distributions discuss the value of probability ideas for statistical inference and its use in life tables construct a life-table and utilise it for various demographic calculations explain approaches to population projections

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To put your footer here go to View > Header and Footer 3 Learning Objectives – this session At the end of this session, you will be able to explain the meaning of probability discuss different approaches used to define probability have an appreciation of the probabilities underlying a life table

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To put your footer here go to View > Header and Footer 4 Probability type statements Some typical statements often heard are … It is highly unlikely that students will all arrive in time for lectures on this module The chance of HIV being eradicated in the next 5 years is nil There is little likelihood that climate change can be stopped It is certain that malaria occurs as a result of mosquito bites

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To put your footer here go to View > Header and Footer 5 Quantifying probability statements Can we quantify the above statements in some way? Try allocating a percentage value to each of the above statements which expresses the degree of belief you have in each statement. Note down your answers alongside slide 4 of this handout. We will consider some of your answers and discuss what this means.

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To put your footer here go to View > Header and Footer 6 What is probability? In very simple terms, probability is a number ranging from 0 to 1 (rather than a percentage from 0% to 100%). Here 0 means that the event is impossible, while 1 represents absolute certainty in the event. Values between 0 and 1 indicate the degree to which the event can be expected to happen.

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To put your footer here go to View > Header and Footer 7 Continuing the class exercise… Statements in slide 4 can also be expressed as questions: 1.How likely is it that you will arrive in time for all lectures on this module? 2.What is the chance of HIV being eradicated in the next 5 years? 3.What is the likelihood that climate change can be stopped 4.What is the chance that malaria occurs as a result of mosquito bites

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To put your footer here go to View > Header and Footer 8 Your task … For each question, give a guess in terms of a probability value between 0 and 1 Write down the number of the question, and your answer to each, on one of the small blank cards provided We will collect the cards, look at the answers and discuss the results of belief in each question by class participants You will be invited to comment on whether you think this is an appropriate way of finding the answers!!

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To put your footer here go to View > Header and Footer 9 Subjective approach … Acceptance of the above approach to finding probabilities depends on consistency between answers and assessing the subjective plausibility of values given This can be regarded as a form of subjective probability, i.e. degree to which a person (or community) believes that a proposition is true

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To put your footer here go to View > Header and Footer 10 Classical approach … Probabilities can also be derived on some occasions using logic, e.g. –Tossing a six-sided die, each side is expected to come up with probability 1/6. –Tossing a coin – if this is a fair coin, expect each side to appear with probability ½. Here, probability is based on assuming there is a set of equally likely outcomes and interest is in the probability of one outcome occurring.

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To put your footer here go to View > Header and Footer 11 Example of the classical approach … Consider a lottery in which 6 balls are drawn without replacement from 49 balls numbered 1-49. The person whose ticket matches the numbers on all the six balls drawn wins (or shares) the jackpot. What is the probability of winning the jackpot? Use the classical assumption that every ball has the same chance of being selected. The probability that your first choice is the first ball drawn is 1 / 49 ; then that your second is the second ball drawn is 1 / 48.

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To put your footer here go to View > Header and Footer 12 Example of the classical approach: 2 Thus, the probability of 6 out of 6 matches is:- 1 / (49x48x47x46x45x44). However, we dont have to put the 6 choices in the correct order: there are 6 ways of choosing the first of our numbers, then 5 of selecting the second etc so overall:- Prob(jackpot) = 6x5x4x3x2x1 / (49x48x47x46x45x44). This is 0.0000000715112 or one chance in 13,983,816.

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To put your footer here go to View > Header and Footer 13 Frequentist approach to probability Question: What is the chance that a new born baby will be a girl? e.g. record at regular intervals, the proportion of girls born at a maternity hospital, giving results (say) as shown. No. of babies No. of girls Proportion of girls 1050.500 50240.480 100490.490 150730.487 200980.490 5002430.486 10004880.488 20009750.488 500024410.488

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To put your footer here go to View > Header and Footer 14 Answer to example: Probability of a girl birth is then the limiting proportion of the ratio no. of girls to total number of births, as the sample size increases, i.e. 0.488 This probability is based on evidence – the approach is referred to as the frequentist approach and became widely accepted from the 19 th century onwards. The subjective approach is also now gaining wide popularity, because of its importance in Bayesian inference.

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To put your footer here go to View > Header and Footer 15 Probability ideas in a Life Table Age range Probability of dying in given age range <10.05465 1-40.01906 5-90.00877 10-140.00604...... 95-990.86743 100+1.00000 Such tables allow surviving numbers to be calculated – useful in population projections for policy and other purposes. Further details are presented in later Sessions.

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To put your footer here go to View > Header and Footer 16 Probability texts (sessions 1-10) A few of many hundreds of books on this material:- Blalock, H.M. (1972) Social Statistics (2nd Edition). McGraw-Hill, London. pp 583. Clarke, G.M. and Cooke, D. (2004). A Basic Course in Statistics. 5th edn. Edward Arnold. Johnson, R.A. and Bhattacharyya, G.K. (2001). Statistics Principles and Methods. 4th edn. Wiley. McClave, J.T. and Sincich T. (2006). A First Course in Statistics. 9th edn. Prentice Hall. Owen, F. and Jones, R. (1990). Statistics. 3rd edn. Pitman Publishing, London, pp 480.

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To put your footer here go to View > Header and Footer 17 Demography texts (Sessions 11-20) Pollard, A.H., Yusuf, F. and Pollard, G.N. (1995) Demographic Techniques, 4th edn. A.C. Wilson, Sydney [previous editions published by Pergamon Press Australia & much-loved by author of sessions] Some of the relatively few good newer books:- Hinde, A. (1998) Demographic Methods. Hodder Arnold, London, U.K. Rowland, D.T. (2003) Demographic Methods and Concepts. Oxford University Press, U.S.A. Siegel, J.S. and Swanson, D.A. (2004) The Methods and Materials of Demography, 2nd edn. Academic Press.

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To put your footer here go to View > Header and Footer 18 Some practical work follows …

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