sadc course in statistics laws of probability (session 02)
TRANSCRIPT
SADC Course in Statistics
Laws of Probability
(Session 02)
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Learning Objectives
At the end of this session you will be able to
• state and explain the fundamental laws of probability
• apply Venn diagrams and the laws of probability to solve basic problems
• explain what is meant by the universal event, union and intersection of events, complement of an event and mutually exclusive events
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Aims of probability sessions
In sessions 3-10, the aim is to:
• build a firm mathematical foundation for the theory of probability
• introduce the laws of probability as a unifying framework for modelling and solving statistical problems
• develop problem solving skills for basic probability type questions
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An Example
A household survey in a certain districtproduced the following information:
• Access to child support grants (yes/no)
• Possession of a birth certificate (yes/no)
• School attendance (yes/no)
The total number of children surveyed was3400.
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Two questions of interest…
• Is a child more likely to get a grant if he/she attends school, or if she/he has a birth certificate?
• What is the probability that a child chosen at random from the surveyed children will attend school, given that he/she does not possess a birth certificate
We will aim to answer these questions below.
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Some survey results• 1750 children have a birth certificate • 850 children have a birth certificate and receive
a child support grant • 1200 children receive a child support grant• 600 children have a birth certificate and receive
a child support grant, but do not attend school• 700 attend school and have a birth certificate
but do not receive a child support grant• 50 children neither go to school nor have a birth
certificate but receive a child support grant• 2450 children attend school
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Answering the questions…
To answer the questions posed in slide 5, it is necessary to determine values for a, b, c, d and e in the graphical representation below.
This diagram is called a Venn diagram.
It is a valuable tool for use in computing probabilities associated with specific events.
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17501200
60050
700
a
bc
d
Birth Certificate
School Attendance
Support grant
e = outside of the three circles
2450
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From survey results (slide 6), we have
(i) a + 600 + b + 700 = 1750
(ii) b + 600 = 850
(iii) 600 + 50 + c + b = 1200
(iv) 700 + b + c + d = 2450
(v) 1750 + 50 + c + d + e = 3400
Class exercise:Determine values for a, b, c, d and e usingthe above equations.
Finding a, b, c, d, e
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Let X, Y, be events that a child gets a grant,given that
(i) he/she has a birth certificate
(ii) he/she attends school.
Let Z be the event that a child attends school, given he/she has no birth certificate.
Then, P(X) = (600 + b)/1750 = 0.49
while P(Y) = (b + c)/(700+b+c+d) = 0.22
Further, P(Z) = (c+d)/(3400–1750) = 0.91
Answers to Questions:
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• The results suggest that access to child support is based more on possession of birth certificate than on school attendance.
• There is a high likelihood that a child will attend school even if he/she does not possess a birth certificate
Conclusions:
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The language of probabilityThe first step towards a good understanding of a culture is to learn the language. In the probability culture, the following terms are commonly used:
• Experiment – any action that can produce an outcome. Try the following experiments and record the outcome: smile to your neighbour, count the number of colleagues with cellphones.
• Sample space – the set of all possible outcomes of an experiment. Denoted by S.
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• Event – any set of outcomes. Thus S is also an event called the Universal event. In the children example, we can define an event E = selecting a child who attends school and receives child support.
• Union – the union of events A and B, written A U B (also A or B), is the event that contains all outcomes in A and outcomes in B.
The shaded area represent the union.
Further definitions
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• Intersection – the intersection of events A and B, written A B (also A and B), is the set of outcomes that belong to both A and B, i.e. it is the overlap of A and B.
The shaded area represents the intersection of the two events A and B.
• Null – or empty set is the event with no outcomes in it. Denoted by Ø.
Definitions continued…
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• Complement – of an event A, denoted by Ac, is the set of outcomes in S which are not in A.
A
S
Ac
The complement of event A is represented by the sky-blue (darker shaded) area.
Definitions continued…
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• Mutually exclusive – also called disjoint events, are events which do not have any outcomes in common. No overlap.
A baby girl A baby boy
Considering the experiment of giving birth, there are two mutually exclusive possible outcomes, either a girl or a boy. Of course we exclude rare events of abnormality.
Definitions continued…
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Fundamental laws of probability
The probability of an event A is a number P(A)which satisfies the following three conditions:
1. 0≤P(A)≤1, i.e. probability is a measure that is restricted between 0 and 1.
2. P(S) = 1, where S is the sample space. That is, the universal set is the sure event.
3. If events A and B are disjoint events, then
P(A U B) = P(A) + P(B).
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Consequences of probability laws
i. P(Ac) = 1 – P(A).
This follows from the fact that S = A U Ac, and because A and its complement are mutually exclusive.
Law 3 implies P(S) = P(A) + P(Ac). Now apply Law 2.
ii. P(Ø) = 0.
This easily follows from (i) since Sc = Ø. There is nothing outside the universe S.
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Consequences (continued)
iii. P(A) = P(A B) + P(A Bc).
This also easily follow from Law 3 because events A B and A Bc are disjoint and together they make up the event A.
iv. P(A U B) = P(A) + P(B) – P(A B)This follows from noting that B and ABc aremutually exclusive, and that their union is AUB.
Hence P(A U B) = P(B) + P(A Bc).
Substituting for P(A Bc) from (iii) abovegives the desired result.
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Sub-events: definition
A is said to be a sub-event of the event B,
if P(A) ≤ P(B), i.e.
If every outcome in A is also an outcome in B.
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Let A be the event that a baby girl is born and B the event that a baby is born.
Hence if A happens we know that B has also happened. However, if B happens we cannot be sure that A has happened.
Thus, the probability of getting a baby girl, in the sample space of all potential mothers, is smaller than the probability of getting a baby!
Sub-events: an example
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Answers to questions in slide 9
Values of a, b, c, d and e are:
a = 200, b = 250, c = 300
d = 1200, e = 100
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