Probability

Ch 6, Principle of BiostatisticsPagano & Gauvreau

Prepared by Yu-Fen Li

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Operations on Events

• a Venn diagram is a useful device for depicting the relationships among events

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A ∩ B“both A and B”

A ∪ B“A or B”

Ac or ,“not A”

Probability

• The numerical value of a probability lies between 0 and 1.

• We have

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The additive rule of probability

The additive rule of probability

• For any two events A and B

– If A and B are disjoint (mutually exclusive)

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𝑃 (𝐴∪𝐵)=𝑃 (𝐴)+𝑃 (𝐵)−𝑃 (𝐴∩𝐵)

𝑃 (𝐴∪𝐵)=𝑃 (𝐴)+𝑃 (𝐵)

The additive rule of probability

• The additive rule can be extended to the cases of three or more mutually exclusive events– If A1, A2, · · · , and An are n mutually exclusive

events, then

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A1

A2A4

A5A3

A8A6

A7

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Joint and Marginal Probabilities

• Joint probability is the probability that two events will occur simultaneously.

• Marginal probability is the probability of the occurrence of the single event.

A1 A2B1 a b a+bB2 c d c+d

a+c b+d n

P(A1)

P(A2B1)

Conditional Probability

• We are often interested in determining the probability that an event B will occur given that we already know the outcome of another event A

• The multiplicative rule of probability states that the probability that two events A and B will both occur is equal to the probability of A multiplied by the probability of B given that A has already occurred

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Independence

• Two events are said to be independent, if the outcome of one event has no effect on the occurrence of the other.– If A and B are independent events,

• •

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Multiplicative rule of probability

• For any events A and B

– If A and B are independent

‘independent’ vs ‘mutually exclusive’

• the terms ‘independent’ and ‘mutually exclusive’ do not mean the same thing.– If A and B are independent and event A occurs,

the outcome of B is not affected, i.e. P(B|A) = P(B). – If A and B are mutually exclusive and event A

occurs, then event B cannot occur, i.e. P(B|A) = 0.

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Bayes’ Theorem

• If A1, A2, · · · , and An are n mutually exclusive and exhaustive events

• Bayes’ theorem states

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exhaustive

mutually exclusive

A1A2

A3

A4A5A6

A7

A8

The Law of Total Probability

• P(A)=P(A1 A2 A3 A4)∪ ∪ ∪ =P(A1) + P(A2 ) + P(A3 ) + P(A4) = 1

• P(B)=P(B∩A1) + P(B∩A2) + P(B∩A3) + P(B∩A4) =P(A1)P(B|A1) + P(A2)P(B|A2) + P(A3)P(B∩A3) + P(A4)P(B|A4)

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B ∩ ∩BB ∩ ∩B

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Examples

• For example, the 163157 persons in the National Health Interview Survey of 1980-1981 (S) were subdivided into three mutually exclusive categories:

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Examples of marginal probabilities

• Find the marginal probabilities

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Example of the additive rule of probability

• If S is the event that an individual in the study is currently employed or currently unemployed or not in the labor force, i.e. S = E1 E∪ 2 E∪ 3.

the additive rule of probability

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Example of the law of total probability

• H may be expressed as the union of three exclusive events:

the law of total probability

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Examples of conditional probabilities

• Looking at each employment status subgroup separately

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Example of Bayes’ theorem

• What is the probability of being current employed given on having hearing impairment?

Diagnostic Tests

• Bayes’ theorem is often employed in issues of diagnostic testing or screening

• Sensitivity and Specificity

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Positive and Negative Predictive Values (PPV and NPV)

• • PPV

• NPV

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Sensitivity (SE)

1-Specificity (1-SP)

A 2 x 2 table

• The diagnostic test is compared against a reference ('gold') standard, and results are tabulated in a 2 x 2 table

Test Gold StandardResults D+ D-

T+a

(TP)b

(FP)

T-c

(FN)d

(TN)

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Sensitivity = a / a+cSpecificity = d / b+d

Positive Predictive Value (PPV) = a / a+b ??

Negative Predictive Value (NPV) = d / c +d ??

Prevalence = a+c / (a+b+c+d) ??

Relationship of Disease Prevalence to Predictive Values

• The probability that he or she has the disease depends on the prevalence of the disease in the population tested and the validity of the test (sensitivity and specificity)

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Example

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D+ D- TotalT+ 20 180 200T- 10 1820 1830

Total 30 2000 2030

30 202030 30

30 20 2000 1802030 30 2030 2000

2000 18202030 2000

2000 1820 30 102030 2000 2030 30

20 / (20 10) 67%

1820 / (180 1820) 91%

20 / (20 180) 10%

1820 / (10 1820) 99.5%

SE

SP

PPV

NPV

( ) 30 / 2030 1.48%P D

Example

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D+ D- TotalT+ 67 9 76T- 33 91 124

Total 100 100 200

67% & 91%

1.48% 67% 6710%

1.48% 67% (1 1.48%) (1 91%) 76

(1 1.48%) 91% 9199.5%

(1 1.48%) 91% 1.48% (1 67%) 124

SE SP

PPV

NPV

( ) 1.48% 50%P D