Topic 6 Probability Modified from the notes of Professor A. Kuk P&G pp

Use letters A, B, C, … to denote events An event may occur or may not occur. Events: passing an exam getting a disease surviving beyond a certain age treatment effective What is the probability of occurrence of an event?

Operations on events 1º Intersection A = “A woman has cervical cancer” B = “Positive Pap smear test” “A woman has cervical cancer and is tested positive”

S A B Venn Diagram

e.g. 6 sided die A=“Roll a 3” B=“Roll a 5” 2° Union

S A B Venn Diagram

“A complement,” denoted by A c, is the event “not A.” A = “live to be 25” A c = “do not live to be 25” = “dead by 25” 3° Complement

S A AcAc Venn Diagram

Null event Cannot happen --- contradiction Definitions:

Cannot happen together: A = “live to be 25” B =“die before 10 th birthday” Mutually exclusive events:

S A B Venn Diagram

Meaning of probability What do we mean when we say P(Head turns up in a coin toss) ? Frequency interpretation of probability Number of tosses Proportion of heads

If an experiment is repeated n times under essentially identical conditions and the event A occurs m times, then as n gets large the ratio approaches the probability of A. More generally, as n gets large

Complement For any event A

Repeat experiment n times A=m A c =n-m Venn Diagram

If A and B are mutually exclusive i.e. cannot occur together Mutually exclusive events

Conduct experiment n times A=m B=k Venn Diagram when A and B are mutually exclusive

If the events A, B, C, …. are mutually exclusive – so at most one of them may occur at any one time – then : Additive Law

A B In general,

Note: Multiplicative rule

Diagnostic tests D = “have disease” D c =“do not have disease” T + =“positive screening result P(T + |D)=sensitivity P(T - | D c )=specificity Note: sensitivity & specificity are properties of the test

PRIOR TO TEST P(D)= prevalence AFTER TEST: For someone tested positive, consider P(D|T + )=positive predictive value. For someone tested negative, consider P(D c |T - )=negative predictive value. Update probability in presence of additional information

D DcDc T+T+

This is called Bayes’ theorem prevalence x sensitivity = prev x sens + (1-prev)x(1-specifity) Using multiplicative rule = positive predictive value = PPV

Example: X-ray screening for tuberculosis 30Total 8Negative 22Positive Yes Tuberculosis X-ray

Example: X-ray screening for tuberculosis Total 17398Negative 5122Positive NoYes Tuberculosis X-ray

Example: X-ray screening for tuberculosis Total 17398Negative 5122Positive NoYes Tuberculosis X-ray

Population: 1,000,000 Screening for TB

Population: 1,000,000 TB: 93 Prevalence = 9.3 per 100,000 No TB: 999,907

Population: 1,000,000 TB: 93 No TB: 999,907 T + 68 T - 25 Sensitivity =

Population: 1,000,000 TB: 93 No TB: 999,907 T + 68 T + 28,497 T - 25 T - 971,410 Specificity =

Population: 1,000,000 TB: 93 No TB: 999,907 T + 68 T + 28,497 T - 25 T - 971,410 T + 28,565 T - 971,435

Population: 1,000,000 TB: 93 No TB: 999,907 T + 68 T + 28,497 T + 28,565 compared with prevalence of

Population: 1,000,000 TB: 93 No TB: 999,907 T - 25 T - 971,410 T - 971,445

Ingelfinger et.al (1983) Biostatistics in Clinical Medicine