Transcript Setting the scene - University of Reading

Introduction to
Probability and
Demography Ideas
(Session 01)
SADC Course in Statistics
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
To put your footer here go to View > Header and Footer
2
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
To put your footer here go to View > Header and Footer
3
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
To put your footer here go to View > Header and Footer
4
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.
To put your footer here go to View > Header and Footer
5
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.
To put your footer here go to View > Header and Footer
6
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
To put your footer here go to View > Header and Footer
7
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!!
To put your footer here go to View > Header and Footer
8
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
To put your footer here go to View > Header and Footer
9
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.
To put your footer here go to View > Header and Footer
10
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.
To put your footer here go to View > Header and Footer
11
Example of the classical approach: 2
Thus, the probability of 6 out of 6 matches is:1/
(49x48x47x46x45x44).
However, we don’t 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.
To put your footer here go to View > Header and Footer
12
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
10
No. of Proportion
girls
of girls
5
0.500
50
24
0.480
100
49
0.490
150
73
0.487
200
98
0.490
500
243
0.486
1000
488
0.488
2000
975
0.488
5000
2441
0.488
To put your footer here go to View > Header and Footer
13
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 19th century onwards.
The subjective approach is also now gaining
wide popularity, because of its importance in
Bayesian inference.
To put your footer here go to View > Header and Footer
14
Probability ideas in a Life Table
Age
range
Probability of dying
in given age range
<1
0.05465
1-4
0.01906
5-9
0.00877
10-14
0.00604
.
.
.
.
.
.
95-99
0.86743
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.
To put your footer here go to View > Header and Footer
15
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.
To put your footer here go to View > Header and Footer
16
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.
To put your footer here go to View > Header and Footer
17
Some practical work follows …
To put your footer here go to View > Header and Footer
18