Topic 6 Probability Modified from the notes of Professor A. Kuk P&G pp. 125-134.

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

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 = =


179030Total

17398Negative

5122Positive

NoYes

Tuberculosis

X-ray

22Sensitivity .7333

3

0 = =


179030Total

17398Negative

5122Positive

NoYes

Tuberculosis

X-ray

1739Speci

22Sensitivi

ficity

ty .7333 3

.9715179

0

0

= =

= =

Population: 1,000,000

Screening for TB


TB: 93

Prevalence= 9.3 per 100,000

No TB: 999,907


TB: 93 No TB: 999,907

T+

68T-

25

Sensitivity

= 0.7333


TB: 93 No TB: 999,907

T+

68T+

28,497T-

25T-

971,410

Specificity0.9715 =


TB: 93 No TB: 999,907

T+

68T+

28,497T-

25T-

971,410

T+

28,565T-

971,435


TB: 93 No TB: 999,907

T+

68T+

28,497

T+

28,565

68PPV =

28,5650.00239=

compared with prevalence of 0.00093


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

Topic 6 Probability Modified from the notes of Professor A. Kuk P&G pp. 125-134.

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Transcript of Topic 6 Probability Modified from the notes of Professor A. Kuk P&G pp. 125-134.