Measuring chance

Probabilities

FETP India

Competency to be gained from this lecture

Apply probabilities to field epidemiology

Key issues

• 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) = 1/2Probabilities

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

Young Old TotalMale 0.48 0.12 0.60Female 0.32 0.08 0.40Total 0.80 0.20 1.00

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• 0.6 + 0.8 = 1.4, absurd result

(Male and Young are counted twice)

• Sex and age are independent• Probability of being male and young

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

Number ofconta-

minatedcultures

Numberof

patients

Proportionatefrequency

0123

364122

131

0.7280.2440.0260.002

Total 500 1.000

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• 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