Topic 6Probability
Modified from the notes of Professor A. Kuk
P&G pp. 125-
134
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”
B" andA " means ,
by denoted ,B" intersectsA "event The
BA
BA
S
A B
Venn Diagram
BA
••
•
•••
• •• •
• ••
• • •••
•••
e.g.6 sided
die
A=“Roll a 3” B=“Roll a 5”
2° Union
5"or 3 a Roll" BA
both"or Bor A either " is
,by denoted ,B"union A "event The BA
S
A B
Venn Diagram
BA
“A complement,” denoted byAc, is the event “not A.”
A = “live to be 25”Ac= “do not live to be 25”
= “dead by 25”
3° Complement
S
A
Ac
Venn Diagram
Null event
Cannot happen --- contradiction
Definitions:
cAAge .,.
Cannot happen together:
A = “live to be 25”B =“die before 10th birthday”
Mutually exclusive events:
BA
S
A
B
Venn Diagram
BA
Meaning of probabilityWhat do we mean when we sayP(Head turns up in a coin toss) ?
21
Frequency interpretation of probability
Number of tosses 10 100 1000 10000Proportion of heads .200 .410 .494 .5017
5.0
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
mn
probability of A.
( )m
P An
=
More generally,
as n gets large
Complement
c
c
mP(A)
nn-m
P(A ) 1 P(A)n
P(A) + P(A ) 1
=
= = -
=
For any event A
1)(0
so ,
AP
nm
Repeat experiment n times
A=m
Ac=n-m
Venn Diagram
If A and B are mutually exclusive
i.e. cannot occur together( )P A B P(A) + P(B)
m+k m kn n n
=
= +
U
Mutually exclusive events
BA
Conduct experiment n times
A=m
B=k
Venn Diagram when A and B are mutually exclusive
If the events A, B, C, …. aremutually exclusive – so at mostone of them may occur at anyone time – then :
Additive Law
)...()()(
...)(
CPBPAP
CBAP
A B
In general, )()()()( BAPBPAPBAP
BA
Note:
Multiplicative rule
)|()()( ABPAPBAP
)(
)()|(
)(
)()|(
BP
BAPBAP
AP
BAPABP
)|()(
)|()()( So
BAPBP
ABPAPBAP
Diagnostic tests
D = “have disease”
Dc =“do not have disease”T+=“positive screening result
P(T+|D)=sensitivity
P(T-| Dc)=specificity
Note: sensitivity & specificity are properties of the test
PRIOR TO TEST
P(D)= prevalenceAFTER TEST:
For someone tested positive, consider
P(D|T+)=positive predictive value.For someone tested negative, consider
P(Dc |T-)=negative predictive value.
Update probability in presence of additional information
D
Dc
T+TD
TDc
)()(
)(
)(
)()|(
TDPTDP
TDP
TP
TDPTDP
c
This is called Bayes’ theorem
prevalence x sensitivity=
prev x sens + (1-prev)x(1-specifity)
Using multiplicative rule
= positive predictive value = PPV
)|()()|()(
)|()()|(
cc DTPDPDTPDP
DTPDPTDP
similarly obtained becan )|(NPV TDP c
Example: X-ray screening for tuberculosis
30Total
8Negative
22Positive
Yes
Tuberculosis
X-ray
22Sensitivity .7333
3
0 = =
Example: X-ray screening for tuberculosis
179030Total
17398Negative
5122Positive
NoYes
Tuberculosis
X-ray
22Sensitivity .7333
3
0 = =
Example: X-ray screening for tuberculosis
179030Total
17398Negative
5122Positive
NoYes
Tuberculosis
X-ray
1739Speci
22Sensitivi
ficity
ty .7333 3
.9715179
0
0
= =
= =
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+
68T-
25
Sensitivity
= 0.7333
Population: 1,000,000
TB: 93 No TB: 999,907
T+
68T+
28,497T-
25T-
971,410
Specificity0.9715 =
Population: 1,000,000
TB: 93 No TB: 999,907
T+
68T+
28,497T-
25T-
971,410
T+
28,565T-
971,435
Population: 1,000,000
TB: 93 No TB: 999,907
T+
68T+
28,497
T+
28,565
68PPV =
28,5650.00239=
compared with prevalence of 0.00093
Population: 1,000,000
TB: 93 No TB: 999,907
T-
25T-
971,410
T-
971,445 999974.0
435,971
410,971NPV
Ingelfinger et.al (1983) Biostatistics in Clinical Medicine
Ingelfinger et.al (1983) Biostatistics in Clinical Medicine