PROBABILITY PROBABILITY n Medicine is an inexact science
PROBABILITY
PROBABILITY n Medicine is an inexact science n Physicians can rarely predict an outcome with absolute certainty. n Results of available diagnostic tests are not necessarily absolutely accurate, but it definitely affect the probability of presence or absence of disease
PROBABILITY n The theory of probability underlies the methods for drawing statistical inferences in medicine through quantifying the uncertainty inherent in the decision –making process
PROBABILITY n Probability theory allows clinicians to draw conclusions about a population of patients based on known information about a sample drawn from that population.
PROBABILITY n It is the likelihood of occurrence of a certain event compared to the total events
PROBABILITY no. of times E occurs n P(E)= ------------------ no. of times E can occur n P(E): Probability of occurrence of event E n The probability of an event is a non-negative number n 0 ≤ P (E) ≤ 1
PROBABILITY Probability values lie between 0 and 1 n A value of 0 means the event can not occur n A value of 1 means the event will definitely occur n A value of 0. 5 means that the probability of occurrence of the event is equal to the probability of non-occurrence of that event n
PROBABILITY n The sum of the probabilities (or relative frequencies) of all event that can occur in the sample must be 1 (or 100%(
Distribution of S. Cholesterol values (mg/dl) of 1047 individuals S. Cholesterol (mg/dl) frequency RF% CRF% <160 31 134 3 12. 8 3 15. 8 358 326 145 43 12 1047 34. 2 31. 1 13. 7 4. 1 1. 2 100 50. 0 81. 1 94. 8 98. 9 100 160 -199 200 -239 240 -279 280 -319 320 -359 360+ total
2 x 2 table Test +ve Test -ve Total Disease +ve -ve Total 7 4 11 (T + ve ) (F + ve ) 3 86 (F - ve ) (T -ve ) 10 90 89 100
2 x 2 table n T+ means the persons have the disease & they show test +ve n T- means the persons have the disease & they show test -ve n F+ (false +ve) means the persons have no disease & they show test +ve n F- (false –ve) means the persons have the disease & they show test –ve
Marginal Probabilities n P(D+) = = 10/100=0. 1 = 10% n P(D-) = =90/100= 0. 9 =90% Test n P(T+) = =11/100 = 0. 11=11% Test P(T-) = =89/100 = 0. 89 =89% Total n Disease Total +ve -ve +ve 7 4 11 3 86 89 10 90 100 -ve
Joint probability The probability of occurrence of two or more events simultaneously n P(A and B) n
Joint probability n P(D+&T+) = =7/100=0. 07 =7% n P(D-&T+) = =4/100= 0. 04 =4% Test n P(T-&D+) = =3/100 = 0. 03=3% Test n Disease Total +ve -ve +ve 7 4 11 3 86 89 10 90 100 -ve P(T-&D-) = Total =86/100 = 0. 86=86%
Conditional probability n The probability of occurrence of an event given that another event had already occurred n P (B I A) = P (B and A) / P(A) n Occurrence of event B when event A had already occurred
Conditional probability n P(D+ I T+) = P(D+&T+) / P(T+) Disease Total +ve -ve Test +ve 7 4 11 3 86 89 10 90 100 =(7/100) / (11/100) Test =7/11 -ve =63. 64% Total
Conditional probability P(D- I T+) =P(D-&T+) / P(T+) Disease Total +ve -ve Test +ve 7 4 11 3 86 89 10 90 100 = (4/100) / (11/100) Test =4/11 -ve =36. 36% Total
Multiplication rules Independent events: If the occurrence of event A is not affected by occurrence of event B n P(A and B) = P(A) X P(B) Multiplication rules of probability (when “ and” is used) n
Multiplication rules / Independent events n What is the probability of selecting two patients randomly , both of them had S. cholesterol level of <160 mg/dl? =0. 03 X 0. 03 = 0. 0009 =0. 09% S. Ch (mg/dl) No. R F% CRF% <160 31 3 3 160 -199 134 12. 8 15. 8 200 -239 358 34. 2 50. 0 240 -279 326 31. 1 81. 1 280 -319 145 13. 7 94. 8 320 -359 43 4. 1 98. 9 360+ 12 1. 2 100 total 1047 100
Multiplication rules/ Independent events n What is the probability of having two boys in two successive pregnancies? = 0. 5 X 0. 5 = 0. 25 =25%
Multiplication rules n Non independent events Occurrence of event A is affected by occurrence of event B (if the two events are related or associated) P (A and B) = P (A I B). P (B) n So it is joint probability
Multiplication rules/ Non independent events What is the probability of selecting an individual who is disease –ve & test –ve? n P(D- and T -) =86/100 = 86% Disease Total +ve -ve Test +ve 7 4 11 Test -ve 3 86 89 10 90 100 Total
Probability rules Addition rules of probabilities When “or” is used n
Addition rules Mutually exclusive events: the events that can not occur together, i. e. : the occurrence of one event will exclude the occurrence of the other event n P(A or B)= P(A) + P(B)
Addition rules/ Mutually exclusive events n n What is the probability of selecting at random a person with serum cholesterol of <160 or > 360 mg/dl P(S. ch<160)+P(S. ch> 360) = 3%+1. 2% = 4. 2% S. Ch (mg/dl) No. R F% CRF% <160 31 3 3 160 -199 134 12. 8 15. 8 200 -239 358 34. 2 50. 0 240 -279 326 31. 1 81. 1 280 -319 145 13. 7 94. 8 320 -359 43 4. 1 98. 9 360+ 12 1. 2 100 total 1047 100
Addition rules n Non mutually exclusive events P(A or B) = P(A)+ P(B) – P(A and B) n
Addition rules/ Non mutually exclusive events n P(D- or T-) = P(D-) + P(T-) – P(D- and T -) Disease +ve -ve Total Test +ve 7 4 11 Test -ve 3 86 89 10 90 100 = (90/100)+ (89/100)(86/100) =93/100 =93% Total
Exercise Results of blood type and sex of 100 O individuals. What is the probability A that an individual picked at random from B this group has: P( Female I B. group A) AB P(B. group A I Female) Total n Male Female Total 20 20 40 17 18 35 8 7 15 5 5 10 50 50 100
Exercise n n n n P(B. group O or A) P(B. Group O and Male) P(B. group AB I Male) P( Male &B. group B) P(Female & B. group O) P( Male or B. group AB) P(B. group B) Male Female Total O 20 20 40 A 17 18 35 B 8 7 15 AB 5 5 10 Total 50 100 50
Exercise ill Not ill Total 90 30 120 Did not eat 20 Fish 60 80 Total 90 200 Ate Fish 110
Exercise n What is the probability that a student becomes ill after eating Fish? n What is the probability that a student does not become ill after eating Fish? n What is the probability that a student becomes ill if no Fish is eaten?
Exercise n What is the probability that a student who attended the party becomes ill? n What is the probability that a student with food poisoning ate Fish? n What is the probability that a student who attended the party did not eat Fish?
Exercise The following table shows the outcome of 500 interviews Outcome completed during a Area of city For(F) Against(Q) Undecided(R) survey to study the opinions of A 100 20 5 125 residents of a B 115 5 5 125 certain city about C 50 60 15 125 legalized abortion. D 35 50 40 125 The data are also classified by the 300 135 65 500 area of the city in which the Calculate the following probabilities: questionnaire was 1 - P(A and R) 2 - P( Q or D) attempted. 3 - P( D) 4 - P( Q/D) 5 - P( B/R)
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