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1. Introduction to Probability

2. Section Goals
After completing this section, you should be able to:
• Explain basic probability concepts and definitions
• Use a Venn diagram to illustrate simple probabilities
• Apply common rules of probability
• Compute conditional probabilities
• Determine whether events are statistically
independent

3. Why Probability?
• A portion of data is normally required to make inference about
the larger population.
But you will never be able to know the true value (population)
(for example if selected at random, what is the chance that you
pick a women with HIV among ANC clients
OR
• You may know the domain of the outcome, but you don’t know
which element will take place (for example if you select one
person at random out of 1000)
Therefore
• The link between the sample and original body of the data is
based on the theory of probability.

4. Why Probability in health?
 Results are not certain. To formulate a diagnosis, a physician must
rely on available diagnostic information about a patient
 History and physical examination
 Laboratory investigation, X-ray findings, ECG, etc

5. Probability…..
More importantly probability theory is used to
understand:
 About probability distributions:
Binomial, Poisson, and Normal
Distributions
 Sampling and sampling distributions
 Estimation
 Hypothesis testing
 Advanced statistical analysis

6. Cont’
 Although no test result is absolutely accurate, it does affect the
probability of the presence or absence of a disease.
 Sensitivity and specificity
• The ability of the screening test to identify correctly
those who have the disease is measured by the
sensitivity of the test.
• The ability of the screening test to identify correctly
those who do not have the diseases is measured by the
specificity of the test.
 An understanding of probability is fundamental for quantifying
the uncertainty that is inherent in the decision-making process.

7. Probability
 Probability - the chance of an event occurring.
 Inferences are made about a population based on a sample
 Probability is viewed as a Measure of Reliability for an Inference.
♦Probability provides a measure of the uncertainty (or
certainty) associated with the occurrence of events or
outcomes
♦Probability is useful in exploring and quantifying
relationships

8. Definition
• Random Experiment – a process leading to an
uncertain outcome.
• Outcome is the result of a single trial of an experiment.
A Head
A four

9. Cont.…
• Sample Space – the collection of all possible
outcomes of a random experiment
• Event – any subset of basic outcomes from the
sample space
• Independent event -The outcome of one event has
no effect on the occurrence or non-occurrence of
the other.

10. Cont.…
• Intersection of Events – If A and B are two events in a
sample space S, then the intersection, A ∩ B, is the
set of all outcomes in S that belong to both A and B
(continued)
A B
AB
S

11. Cont.…
• A and B are Mutually Exclusive Events if they have no
basic outcomes in common
• i.e., the set A ∩ B is empty
• Weight of an individual can’t be classified simultaneously
as “underweight”, “normal”, “overweight”
A B
S

12. Cont.…
• Union of Events – If A and B are two events in a
sample space S, then the union, A U B, is the set of
all outcomes in S that belong to either A or B
(continued)
A B
The entire shaded area
represents
A U B
S

13. Cont.…
• The Complement of an event A is the set of all basic
outcomes in the sample space that do not belong to
A. The complement is denoted
(continued)
A
A
S
A

14. Examples
Let the Sample Space be the collection of all
possible outcomes of rolling one die:
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then
A = [2, 4, 6] and B = [4, 5, 6]

15. (continued)
Examples
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
5]
3,
[1,
A 
6]
[4,
B
A 

6]
5,
4,
[2,
B
A 

S
6]
5,
4,
3,
2,
[1,
A
A 


Complements:
Intersections:
Unions:
[5]
B
A 

3]
2,
[1,
B 

16. • Mutually exclusive:
• A and B are not mutually exclusive
• The outcomes 4 and 6 are common to both
• Collectively exhaustive:
• A and B are not collectively exhaustive
• A U B does not contain 1 or 3
(continued)
Examples
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

17. Counting Rules
The Addition Rule
 If A ∩ B = Ø, then n(A ∪ B) = n(A) + n(B)
 If A1, A2, . . . , Ak are k pair-wise mutually exclusive events,
then n(A1∪A2 ∪ · · · ∪Ak ) =∑ n(Ai)
 But, For any events A & B,
n (A ∪ B)=n (A) + n (B) – n(A∩ B).

18. The Multiplication Rule
 Rule1
 If each event in a sequence of n events has K possibilities,
then the total number of possibilities will be. K.… K = Kn
 Example
Seven dice are rolled. How many different outcomes are
there?
K = 6; n = 7 Thus, Total = 67

19. The Multiplication Rule …
Rule-2
• In a sequence of n events, if there are m ways a first
event can occur and n ways a second event can
occur, the total number of ways the two events can
occur is given by m x n.

20. The Multiplication Rule …
Example
There are 8 different Biostatistics, 6 different Epidemiology and
3 different Nursing books. A student must select one book of
each type. How many different ways can this be done?
K1 = 8; K2 = 6; K3 = 3
Total = 8 x 6 x 3 = 144

21. Permutation
 An arrangement of n objects in a specific order.
• Factorial: n! = n x (n – 1) x (n – 2) x ... x 1
Note that 1! = 0! = 1 by definition.
 The number of permutation of n objects taken all together
is given by nPn (read as n permutation n) = n!
 An arrangement of n objects in a specific order using r
objects at a time is given by:
r)!
(n
n!
r
nP



22. Permutation . . .
 Example
1. Suppose that a photographer must arrange three people in
a row for a photograph. How many different possible ways
can the arrangement be done?
n = 3
Since the photo is going to be taken all together, the total possibility is given
by: 3P3 = 3! = 3 x 2 x 1 = 6

23. Permutation . . .
2. How many different four – letter permutations can be
formed from the letters in the word DECAGON?
n = 7; r = 4
Total number of ways = 7P4 = 7!/ (7-4)! = 7x6x5x4 = 840

24. Combination
 A selection of objects without regard to order.
 Example: Given the letters A, B, C and D. List the
permutations and combinations for selecting two letters.
 The number of combination of r objects selected from n
objects is
r!
r
nP
r!
r)!
(n
n!
r
nC 


Permutation AB; AC; AD; BA;
BC; BD; CA; CB;
CD; DA; DB; DC
Combination AB; AC; AD; BC; BD;
CD

25. Combination . . .
Example
1. Suppose the donor plan to invest equal amounts of money in
each of five hospitals. If there are 20 hospitals from which to
make the selection, how many different samples of five
hospitals can be selected from the 20?
n = 20; r = 5;
Total = 20C5 = 20!/ (20-5)!x5!
= 20!/ 15!x5!
= 15, 504

26. Exercise
• How many ways can a jury of 6 men and 4 women be selected from
10 men and 8 women?

27. Probability
• Probability – the chance that an
uncertain event will occur
(always between 0 and 1)
0 ≤ P(A) ≤ 1 For any event A
Certain
Impossible
.5
1
0

28. Assessing Probability
• There are three approaches to assessing the probability
of an uncertain event:
1. classical probability
• Assumes all outcomes in the sample space are equally likely to occur
• Example: what is the probability of male for the newly delivered child
at standard condition?
• P(m)=0.5
space
sample
the
in
outcomes
of
number
total
event
the
satisfy
that
outcomes
of
number
N
N
A
event
of
y
probabilit A



29. Assessing Probability
2. Relative frequency probability
• the limit of the proportion of times that an event A occurs in a large number of trials, n
Example: for the previous example if 3 women delivered children, what will be the
probability of all of the children’s sex is male?
P(MMM)=0.125
3. Subjective probability
an individual opinion or belief about the probability of occurrence
population
the
in
events
of
number
total
A
event
satisfy
that
population
the
in
events
of
number
n
n
A
event
of
y
probabilit A



30. Probability Rules
• The Complement rule:
• The Addition rule:
• The probability of the union of two events is
1
)
A
P(
P(A)
i.e., 

P(A)
1
)
A
P( 

B)
P(A
P(B)
P(A)
B)
P(A 





31. • It is concerned with the probability of a union of
outcomes.
 i.e., for two events A and B: simple probability that A occurs
added to the simple probability that B occurs.
 If events A and B are mutually exclusive, the probability
that A or B occurs is given by P (A U B) = P (A) + P (B)
Example
A day of the week is selected at random. Find the
probability that it is weekend day {Saturday or Sunday}.
S = { Mo, Tu, We, Th, Fr, Sa, Su}
A = {Sa}; B = {Su}
P(A) = 1/7; P(B) = 1/7
P(A u B) = P(A) + P(B) = 1/7 + 1/7

32. The Addition . . .
If two events A and B are not mutually exclusive, then, P (A
U B) = P (A) + P (B) – P (A and B)
Example
1. There are 80 nurses and 40 physicians in a hospital. Of
these, 70 nurses and 15 physicians are females. If a staff
person is selected at random, find the probability that the
subject is a nurse or male.
P(N u M) = P(N) + P(M) – P(N n M)
= 80/120 + 35/ 120 – 10/ 120 = 105/ 120
Male Female Total
Nurse 70
Physician 25 15 40
Total 85 120
80
35
10

33. A Probability Table
B
A
A
B
)
B
P(A 
)
B
A
P( 
B)
A
P( 
P(A)
B)
P(A 
)
A
P(
)
B
P(
P(B) 1.0
P(S) 
Probabilities and joint probabilities for two events A and
B are summarized in this table:

34. Conditional Probability
• Refers to the probability of an event, given that
another event is known to have occurred.
• When thinking about conditional probabilities, think in
stages. Think of the two events A and B occurring
chronologically, one after the other, either in time or
space.
• The conditional probability of an event A given an
event B is present is:
• P(A|B)= P(A ∩B)/P(B), where P(B)≠0

35. P(B)
B)
P(A
B)
|
P(A


P(A)
B)
P(A
A)
|
P(B


The conditional
probability of A given
that B has occurred
The conditional
probability of B given
that A has occurred
Conditional Probability

36. • What is the probability that a patient is diabetic,
given that he is hypertensive?
i.e., we want to find P(D | H)
Conditional Probability Example
 Of the patients in a given hospital, 70% have
developed hypertension and 40% have developed
diabetes. 20% of the patients have developed both.

37. Conditional Probability Example
No D
D Total
H 0.2 0.5 0.7
No H 0.2 0.1 0.3
Total 0.4 0.6 1.0
 .
(continued)

38. Conditional......
Example:
A study investigating the effect of prolonged exposure to bright light
on retina damage in premature infants.
Retinopathy
YES
Retinopathy
NO
TOTAL
Bright light
Reduced light
18
21
3
18
21
39
TOTAL 39 21 60

39. Conditional......
 The probability of developing retinopathy is:
P (Retinopathy) = No. of infants with retinopathy
Total No. of infants
= (18+21)/(21+39)
= 0.65

40. Conditional......
 We want to compare the probability of retinopathy, given that the
infant was exposed to bright light, with that the infant was exposed to
reduced light.
 Exposure to bright light and exposure to reduced light are
conditioning events, events we want to take into account when
calculating conditional probabilities.

41. Conditional.......
 The conditional probability of retinopathy, given exposure
to bright light, is:
P(Retinopathy/exposure to bright light) =
No. of infants with retinopathy exposed to bright light
No. of infants exposed to bright light
= 18/21 = 0.86

42. Conditional....
P(Retinopathy/exposure to reduced light) =
# of infants with retinopathy exposed to reduced light
No. of infants exposed to reduced light
= 21/39 = 0.54
 The conditional probabilities suggest that premature infants
exposed to bright light have a higher risk of retinopathy than
premature infants exposed to reduced light.

43. •Marginal probabilities can be calculated:
•P(Male) =
•P(Young) =
•Joint probability can be calculated:
•P(Female and Older )=
•Using the addition rule of probability:
•P( Female or Older)= P( Female) + P(Older) – P(Female and Older)=

44. Calculating Conditional Probabilities
• P(Older given Male) = P(Older | Male) = 0.40
• P(Older given Female) = P(Older | Female) = 0.73
• Comparing Conditional Probabilities
– For males: P(Older | Male) = 0.40
– For females: P(Older | Female) = 0.73
– For all patients :P(Older) = 0.65

45. Are Characteristics Sex & Age independent?
• If sex and age are independent: Then the probability of
being in a particular age group should be the same for
both sexes
• In other words, the conditional probabilities should be
equal
• Assess whether:
• P(Older | Male) = P(Older |Female) = P(Older)

46. Multiplication Rule
• Multiplication rule for two events A and B:
• also
P(B)
B)
|
P(A
B)
P(A 

P(A)
A)
|
P(B
B)
P(A 


47. Statistical Independence
• Two events are statistically independent if and only
if:
• Events A and B are independent when the probability of one event is
not affected by the other event
• If A and B are independent, then
P(A)
B)
|
P(A 
P(B)
P(A)
B)
P(A 

P(B)
A)
|
P(B 
if P(B)>0
if P(A)>0

48. Statistical Independence Example
No D
D Total
H 0.2 0.5 0.7
No H 0.2 0.1 0.3
Total 0.4 0.6 1.0
Are the two events Hypertensive and Diabetic statistically independent?

49. Statistical Independence Example
No D
D Total
H 0.2 0.5 0.7
No H 0.2 0.1 0.3
Total 0.4 0.6 1.0
P(H ∩ D) = 0.2
P(H) = 0.7
P(D) = 0.4
P(H)P(D) = (0.7)(0.4) = 0.28
P(H ∩ D) = 0.2 ≠ P(H)P(D) = 0.28
So the two events are not statistically independent

50. Bayes Theorem

51. Bayes Theorem

52. Screening tests, Sensitivity and specificity
• In the health science field a widely used application of
probability laws and concepts is found in the evaluation of
screening tests and diagnostic criteria.
• Of interest to clinicians is an enhanced ability to correctly
predict the presents or absence of the disease from a
knowledge of a test result. Testing procedure may yield a
false positive or a false negative.
• A false positive results when a test indicates a positive
status when the true status is negative.
• A false negative results when a test indicates a negative
status when the true status is positive.

53. Screening tests, Sensitivity and specificity
• We may compute the conditional probability estimate p(T/D)=a/a+c
this ratio is an estimate of sensitivity of screening test.
• The sensitivity of a test (or symptom) is the probability of a positive
test result (or presence of the symptom) given the presence of the
disease.
• The specificity of a test (or symptom) is the probability of a negative
test result (or absence of the symptom) given the absence of the
disease. i.e P(NT/ND)
Disease
Present (D) Absent (ND) Total
Positive(T) a b a+b
Negative(NT) c d c+d
Total a+c b+d n

54. Summary
• Probabilities can describe certainty associated with an event or
characteristic
• Types of events:
– Mutually exclusive events
– Independent events
• Addition and multiple rules of probability
• Types of probabilities:
– Marginal,
– Joint,
– Conditional
• Applications of probability (e.g. screening tests)