Transcript Getting to the essential

Measuring chance
Probabilities
FETP India
Competency to be gained
from this lecture
Apply probabilities to field epidemiology
Key issues
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Probabilities
Rule of addition
Rule of multiplication
Non-independent events
Question 1
• Suppose:
 The success rate of a programme to stop smoking
is 75% compared to the expected 70%
• Can we really be certain that the programme
is successful?
Probabilities
Question 2
• Suppose:
 The mean height of 200 adults in a suburban
area of a city is 165 cm compared to the city’s
mean height of 170 cm
• Can we really be certain that people in the
suburb have a shorter height?
Probabilities
Question 3
• Suppose:
 In a trial involving 100 patients, treatment A is
better than treatment B
• Can we really be certain that treatment A is
better than treatment B?
Probabilities
Probabilities and statistical inference
• In statistics, we infer from a sample to a
population
• In that process, there is a element of chance
• We study probabilities (science) to measure
this element of chance (intuitive notion)
Probabilities
Tossing a coin
• Two possible events (Outcomes):
 Head or tail
• Probability of getting a head in a toss:
 1/2 = 50%
• Probability of getting a tail in a toss:
 1/2 = 50%
Probabilities
Throwing a dice
• Six possible events (Outcomes):
 1,2,3,4,5 or 6
• Probability of getting a score of 1:
 1/6
• Probability of getting a score of 4:
 1/6
• Probability of getting a score of 6:
 1/6
Probabilities
Drawing a card from a pack
• There are 52 cards in a pack of playing cards
which includes:
 4 aces, 2 red and 2 black
• A card is randomly picked from the pack:
 Probability of getting an ace: 4/52
 Probability of getting a black ace: 2/52
 Probability of getting a red ace: 2/52
Probabilities
Using experience as a relative frequency
• Suppose a coin is tossed 10,000 times and
head (H) has occurred 4,980 times
• The relative frequency of head is:
 H = 4,980  10,000 = 0.498  0.5
Probabilities
Theoretical approach
• Assuming that the coin is fair, both head (H)
and tail (T) have equal chance of occurring
• Probability of the event “head”:
Number of outcomes of interest (say, Head)
Number of possible outcomes
i.e., P(H) = 1/2
and P(T) = 1/2
Probabilities
Generic concept of probabilities
• Numerator
 Event of interest
• Denominator
 All the possible events that may occur
• Probabilities are proportions:
 They range between 0 and 1
 The numerator is part of the denominator
Probabilities
Definition of probabilities
• Probability is defined as a proportionate
frequency
• If a variable can take any of N values and n
of these constitute the event of interest to
us, the probability of the event is given by
n/N
Number of outcomes of interest
Total number of outcomes
Probabilities
Rule of addition
• Mutually exclusive events
 P(A) = Probability of event A occurring
 P(B) = Probability of event B occurring
• The two events A and B are said to be mutually
exclusive if they cannot occur together
• In this case, the probability that one OR the other
occurs is the sum of the two individual probabilities
 P(A or B) = P(A) + P(B)
Rule of addition
“OR”: Additive probabilities
• What is the probability of getting a 3 (1/6) OR a 5
from a dice (1/6)
 Probability of getting a score of 3 = 1/6
 Probability of getting a score of 5 = 1/6
• 3 and 5 cannot occur at the same throw
• The total number of possible events remains 6
• “1 OR 6” constitute 2 of the 6 possible events
 Probability: 2 / 6
• The probability of getting an event or the other is
the sum of the individual probabilities
 1/6 + 1/6 = 2/6
Rule of addition
Additive probabilities
• When the events are mutually exclusive and
collectively exhaustive, the probability of
each event add up to 1
• Probability of getting a head in a coin toss:
 0.5
• Probability of getting a tail in a coin toss:
 0.5
• Probability of getting a head OR a tail
 0.5 + 0.5 = 1
Rule of addition
Rule of multiplication
• P(A) = Probability of an event A occurring
• P(B) = Probability of an event B occurring
• The two events A and B are said to be
independent if the occurrence of one has no
implications on the other
• In this case, the probability of both A and B
occurring at the same time is the product of
the two individual probabilities
 P (AB) = P (A) x P (B)
Rule of multiplication
“AND”: Multiplicative probabilities
• A coin is tossed and a dice is thrown simultaneously
• The outcome of the toss of the coin has no
implication on the result of the throw of the dice
• What is the probability of getting a head from a coin
(1/2) AND a 6 from a dice (1/6)
• The total number of possible events is a
multiplication of the possible events
 2 x 6 = 12
• “Head AND 6” is only one of the 12 possible events
 Probability: 1 / 12
• The probability of getting a combination of events is
a multiplication of the individual probabilities
 1/6 x 1/2 = 1/12
Properties of the events considered
so far
• Mutually exclusive
 If the tossed coin shows head, it does not shows
tail
• Independent
 The outcome of the coin tossing does not
influence the dice throwing
Rule of multiplication
Considering non-independent events
• Village survey
• Event A:
 Being female
 P (A): Probability of being female
• Event B:
 Being under 5
 P (B): Probability of being under 5
• Event A and event B are not independent
Non independent events
Change in the additive rule in the case of
non-mutually exclusive events
• If the events are not mutually exclusive, the total
probabilities exceed one
• Probability of being female, P(A) also includes
female under 5
• Probability of being under 5, P(B) also includes
female under 5
• Female under 5 are counted twice
• Subtract the probability of the combined events
 P(A OR B) = P(A) + P (B) - P (A AND B)
Non independent events
In
Example from a clinical trial
• Proportion of male patients = 0.60
• Proportion of young patients = 0.80
• We wish to determine the probability of patients
who were either male or young or both
Young Old
• 0.6 + 0.8 = 1.4, absurd result
Male
0.48
0.12
 (Male and Young are counted twice)
Female 0.32
Total
0.80
• Sex and age are independent
• Probability of being male and young
Total
0.60
0.08 0.40
0.20 1.00
 0.6 x 0.8 = 0.48
• Proportion who are either male or young ( or both)
 0.6 + 0.8 - 0.48 = 0.92
Non independent events
Change in the multiplicative rule in the
case of non-independent events:
Conditional probabilities
• The probability of being a female under 5 is
not equal to P (A) x P (B)
• P (A AND B) = P (A) x P (B, given A)
= P (B) x P (A, given B)
• P (B, given A) is the probability of getting
the event “under 5” (B) GIVEN that the
event “female” occurred (A)
Non independent events
Selection of a subject in a survey
• Survey in a small community of 800 subjects
 128 are aged under 5 years of age
 192 are 5–15 years of age
 480 are aged above 15 years
• A subject is selected at random
• Probability of selecting a child under 5 years of age
 128 / 800 = 0.16
• Probability of selecting a child 5 to 15 years of age
 192 / 800 = 0.24
• Probability of selecting a child older than 15 years
 128+192 / 800 = 320 / 800 = 0.40
Non independent events
Illustration of conditional probabilities
• Consider a group of 5 persons
 3 males (M1, M2, M3) and 2 females (F1, F2)
• One person is selected at random and then a second
is selected again at random from the remaining 4
• What is the probability of selecting a male twice?
• First round:
 Probability of selecting a male = 3 / 5 (M2)
• Second round:
 There are 4 persons left (M1, M3, F1, F2)
 Probability of selecting a male = 2 / 4
• Probability of selecting a male on both occasions
• 3 / 5 x 2 / 4 = 6 / 20
Non independent events
Checking from first principles
• The first person can be selected in 5 ways
 M1 or M2 or M3 or F1 or F2
• With each of these the second person can be
selected in 4 ways
 (e.g., M1 or M3 or F1 or F2 following M2)
• Total number of ways to select 2 persons
 5 x 4 = 20
• Selection of a male
 First round: 3 ways
 Second round: 2 ways
• Total number of ways to select a male twice
 3 x 2 = 6, required Probability = 6 / 20
Non independent events
Laboratory example
• Probability of ‘0’
contaminations
 0.728
• Probability of getting at
least 2 contaminations
 Probability of getting 2
contaminated cultures +
 Probability of getting 3
contaminated cultures
Number of
contaminated
cultures
0
1
2
3
Total
Number Proportionate
of
frequency
patients
364
122
13
1
500
0.728
0.244
0.026
0.002
1.000
• 0.026 + 0.002 = 0.028
Non independent events
Random variables and probability
distributions
• Statistical experiment is any process by which an
observation (or measurement) is obtained
 Counting the number of sick patients
 Measuring the birth weight of infants
• Variable is called random variable
 Discrete random variable
• The observation can take only a finite number of values
 Continuous random variable
• The observation can take infinite number of values
• The probability distribution is simply an assignment
of probabilities:
 To the specific values of the random variable
 To a range of values of the random variable Non independent events
Key messages
• Probabilities quantify chance
• The probability of occurrence of one or
another mutually exclusive events are added
• The probability of occurrence of one and
another independent event are multiplied
• Non-independent events are addressed
through conditional probabilities