Investopedia Example Bayes Theorem(1)

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TreePlan Student License, For Education Only TreePlan.com Event Disease Disease (A) A No Disease (not A) not A Screening Test Test positive (B) B Test negative (not B) not B Given (enter as pr P(B | A) P(not B | not A) P(A) Solution Probability P(A | B) 0.000% P(not A | B) 100.000% P(A | not B) #DIV/0! P(not A | not B) #DIV/0! P(B) 100.00% P(not B) #DIV/0! PRIOR PROBABIL Enter data into the gray boxes. The contingency table, probabi
TreePlan Student License, For Education Only TreePlan.com A not A POSTERIOR PROB B not B MULTIPLICATION RULE: P(T P(T+) = P
TreePlan Student License, For Education Only TreePlan.com MULTIPLICATION We ha P(D+|
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TreePlan Student License, For Education Only TreePlan.com roportions) True Positive Rate=Sensitivity True Negative Rate=Specificity Prevalence ns Positive Predictive Value (PPV) Negative Predictive Value (NPV) Test Positive Rate Test Negative Rate LITIES AFTER THE NEW INFORMATION IS O B P(A) ility tree and conditional probabilities will autopop PROBABILITY TREES To use this Calculator Step 1: Change the labels for the to be predicted (Event A) in the g Step 3: Enter the required gray cells. The contingency and Solutions table w Step 2: Change the labels for the test (Event B) in the gray ce
TreePlan Student License, For Education Only TreePlan.com 0 not B B P(not A) 1 not B BABILITIES AFTER THE NEW INFORMATION IS OBT A P(B) 1 not A A P(not B) 0 not A T+|D+) * P(D+) = P(D+ and T+) = P(D+|T+)* P(T+) P(D+ and T+) + P(D- and T+)
TreePlan Student License, For Education Only TreePlan.com N RULE: P(D+ and T+) = P(D+|T+)* P(T+) ave P(D+ and T+) and P(T+) Solve for P(D+|T+) |T+)= P(D+ and T+) ÷ P(T+)
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TreePlan Student License, For Education Only TreePlan.com OBTAINED (GIVEN IN THIS CASE) Prior Conditional Probabilities 0 P(B | A) pulate. e outcome gray cells. d probabilities in the the y table, Probability Trees, will autopopulate. screening ells.
TreePlan Student License, For Education Only TreePlan.com 1 P(not B | A) 1 P(B | not A) 0 P(not B | not A) TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(A | B) 0.0000 P(not A | B) 1.0000 P(A | not B) #DIV/0! P(not A | not B) #DIV/0!
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TreePlan Student License, For Education Only TreePlan.com Event B B not B TOTALS B not B TOTALS Joint Probabilities P(A and B) 0 Event B: Test Result Exam Suppo What Suppo Does t
TreePlan Student License, For Education Only TreePlan.com P(A and not B) 0 P(not A and B) 1 P(not A and not B) 0 Joint Probabilities P(A and B) 0 P(not A and B) 1 P(A and not B) #DIV/0! P(not A and not B) #DIV/0!
TreePlan Student License, For Education Only TreePlan.com Ev A P(A and B) P(A and not B) P(A) Event A: Co A 0 0 0 CONTINGENCY TAB mple: ose you randomly selected a card from a d is the probability of selecting a King from ose that you are provided with additional i this new information influence your estim
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TreePlan Student License, For Education Only TreePlan.com vent A not A TOTALS P(not A and B) P(B) P(not A and not B) P(not B) P(not A) 1 ondition Status not A TOTALS 1 1 0 0 1 1 BLE deck of cards (traditional 52 card deck). a card randomly selected from a deck of cards? information that the selected card is a face card. mate of the probability that you selected a King? If so, how? Diag abse typicall the test which i
TreePlan Student License, For Education Only TreePlan.com
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TreePlan Student License, For Education Only TreePlan.com gnostic Testing: When we discuss diagnostic testing in medicine, our events are ence/presence of disease and positive or negative test result. Each company will ly publish a sensitivity and specificity which are characteristics of the test and once is performed, patients are generally counseled based on the PPV and NPV of a test is based on the prevalence of the disease. The calculations are still the same as any other screening test, we have discussed thus far. Try the example below.
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TreePlan Student License, For Education Only TreePlan.com ID 0 1 2 3 4 5 6
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TreePlan Student License, For Education Only TreePlan.com Name Value Prob Pred Kind NS TreePlan 0 0 0 E 2 0 E 2 0 E 2 1 T 0 1 T 0 2 T 0 2 T 0
TreePlan Student License, For Education Only TreePlan.com S1 S2 S3 S4 S5 Row 1 2 0 0 0 9 3 4 0 0 0 4 5 6 0 0 0 14 0 0 0 0 0 2 0 0 0 0 0 7 0 0 0 0 0 12 0 0 0 0 0 17
TreePlan Student License, For Education Only TreePlan.com Col Mark 1 1 5 1 5 1 9 1 9 1 9 1 9 1
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TreePlan Student License, For Education Only TreePlan.com Event Disease Disease (A) King No Disease (not A) not King Screening Test Test positive (B) Face Card Test negative (not B) not a Face Card Given (enter as pr P(Face Card | King) 1 P(not a Face Card | not King) 0.8333333333 P(King) 0.0769230769 Solution Probability P(King | Face Card) 33.333% P(not King | Face Card) 66.667% P(King | not a Face Card) 0.000% P(not King | not a Face Card) 100.000% P(Face Card) 23.08% P(not a Face Card) 76.92% PRIOR PROBABIL Enter data into the gray boxes. The contingency table, probab
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TreePlan Student License, For Education Only TreePlan.com King not King POSTERIOR PROB Face Card not a Face Card MULTIPLICATION RULE: P(T P(T+) = P
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TreePlan Student License, For Education Only TreePlan.com MULTIPLICATION We ha P(D+|
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TreePlan Student License, For Education Only TreePlan.com d roportions) True Positive Rate=Sensitivity True Negative Rate=Specificity Prevalence ns Positive Predictive Value (PPV) Negative Predictive Value (NPV) Test Positive Rate Test Negative Rate LITIES AFTER THE NEW INFORMATION IS O Face Card P(King) bility tree and conditional probabilities will autopop PROBABILITY TREES To use this Calculator Step 1: Change the labels for the to be predicted (Event A) in the g Step 3: Enter the required gray cells. The contingency and Solutions table w Step 2: Change the labels for the test (Event B) in the gray ce
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TreePlan Student License, For Education Only TreePlan.com 0.07692307692 not a Face Card Face Card P(not King) 0.92307692308 not a Face Card BABILITIES AFTER THE NEW INFORMATION IS OBT King P(Face Card) 0.23076923077 not King King P(not a Face Card) 0.76923076923 not King T+|D+) * P(D+) = P(D+ and T+) = P(D+|T+)* P(T+) P(D+ and T+) + P(D- and T+)
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TreePlan Student License, For Education Only TreePlan.com N RULE: P(D+ and T+) = P(D+|T+)* P(T+) ave P(D+ and T+) and P(T+) Solve for P(D+|T+) |T+)= P(D+ and T+) ÷ P(T+)
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TreePlan Student License, For Education Only TreePlan.com OBTAINED (GIVEN IN THIS CASE) Prior Conditional Probabilities 1 P(Face Card | King) pulate. e outcome gray cells. d probabilities in the the table, Probability Trees, will autopopulate. screening ells.
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TreePlan Student License, For Education Only TreePlan.com 0 P(not a Face Card | King) 0.166666666666667 P(Face Card | not King) 0.833333333333333 P(not a Face Card | not King) TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(King | Face Card) 0.3333 P(not King | Face Card) 0.6667 P(King | not a Face Card) 0.00000 P(not King | not a Face Card) 1.00000
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TreePlan Student License, For Education Only TreePlan.com Event B Face Card not a Face Card TOTALS Face Card not a Face Card TOTALS Joint Probabilities P(King and Face Card) 0.076923076923077 Event B: Test Result Example: Suppose yo What is the Suppose th Does this n Solution: P(King)= 4 P(Face Car P(not a fac
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TreePlan Student License, For Education Only TreePlan.com P(King and not a Face Card) 0 P(not King and Face Card) 0.153846153846154 P(not King and not a Face Card) 0.769230769230769 Joint Probabilities P(King and Face Card) 0.076923076923077 P(not King and Face Card) 0.153846153846154 P(King and not a Face Card) 0 P(not King and not a Face Card) 0.769230769230769 P(not a fac As a result estimate o 7.69% to t
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TreePlan Student License, For Education Only TreePlan.com Ev King P(King and Face Card) P(King and not a Face Card) P(King) Event A: Co King 0.076923076923077 0 0.076923076923077 CONTINGENCY TAB ou randomly selected a card from a deck o e probability of selecting a King from a card hat you are provided with additional inform new information influence your estimate o 4/52 =7.69% (see red box) rd|King)= 4/4 =100% (see red box) ce Card | not a King) = 40/48 = 83.33%
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TreePlan Student License, For Education Only TreePlan.com ce Card | not a King) = 40/48 = 83.33% t, the P(King|Face Card) is 4/12 (see gr of the probability that I selected king. T the posterior probability of 33.33% bas
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TreePlan Student License, For Education Only TreePlan.com vent A not King TOTALS P(not King and Face Card) P(Face Card) P(not King and not a Face Card) P(not a Face Card) P(not King) 1 ondition Status not King TOTALS 0.153846153846154 0.230769230769231 0.769230769230769 0.769230769230769 0.923076923076923 1 BLE of cards (traditional 52 card deck). d randomly selected from a deck of cards? mation that the selected card is a face card. of the probability that you selected a King? If so, how? Diag abse typically the test which is % (see blue box)
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TreePlan Student License, For Education Only TreePlan.com % (see blue box) reen box) or 33.33%. Yes, the new information influenced the The probability increased from the apriori probability of sed on the additional information.
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TreePlan Student License, For Education Only TreePlan.com gnostic Testing: When we discuss diagnostic testing in medicine, our events are ence/presence of disease and positive or negative test result. Each company will y publish a sensitivity and specificity which are characteristics of the test and once is performed, patients are generally counseled based on the PPV and NPV of a test s based on the prevalence of the disease. The calculations are still the same as any other screening test, we have discussed thus far. Try the example below.
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TreePlan Student License, For Education Only TreePlan.com e
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TreePlan Student License, For Education Only TreePlan.com ID 0 1 2 3 4 5 6
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TreePlan Student License, For Education Only TreePlan.com Name Value Prob Pred Kind NS TreePlan 0 0 0 E 2 0 E 2 0 E 2 1 T 0 1 T 0 2 T 0 2 T 0
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TreePlan Student License, For Education Only TreePlan.com S1 S2 S3 S4 S5 Row 1 2 0 0 0 9 3 4 0 0 0 4 5 6 0 0 0 14 0 0 0 0 0 2 0 0 0 0 0 7 0 0 0 0 0 12 0 0 0 0 0 17
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TreePlan Student License, For Education Only TreePlan.com Col Mark 1 1 5 1 5 1 9 1 9 1 9 1 9 1
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TreePlan Student License, For Education Only TreePlan.com Event Disease Disease (A) A No Disease (not A) not A Screening Test Test positive (B) B Test negative (not B) not B Given (enter as pr P(B | A) P(not B | not A) P(A) Solution Probability P(A | B) 0.000% P(not A | B) 100.000% P(A | not B) #DIV/0! P(not A | not B) #DIV/0! P(B) 100.00% P(not B) #DIV/0! PRIOR PROBABIL Enter data into the gray boxes. The contingency table, probab
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TreePlan Student License, For Education Only TreePlan.com A not A POSTERIOR PROB B not B MULTIPLICATION RULE: P(T P(T+) = P
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TreePlan Student License, For Education Only TreePlan.com MULTIPLICATION We ha P(D+|
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TreePlan Student License, For Education Only TreePlan.com roportions) True Positive Rate=Sensitivity True Negative Rate=Specificity Prevalence ns Positive Predictive Value (PPV) Negative Predictive Value (NPV) Test Positive Rate Test Negative Rate LITIES AFTER THE NEW INFORMATION IS O B P(A) bility tree and conditional probabilities will autopop PROBABILITY TREES To use this Calculator Step 1: Change the labels for the to be predicted (Event A) in the g Step 3: Enter the required gray cells. The contingency and Solutions table w Step 2: Change the labels for the test (Event B) in the gray ce
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TreePlan Student License, For Education Only TreePlan.com 0 not B B P(not A) 1 not B BABILITIES AFTER THE NEW INFORMATION IS OBT A P(B) 1 not A A P(not B) 0 not A T+|D+) * P(D+) = P(D+ and T+) = P(D+|T+)* P(T+) P(D+ and T+) + P(D- and T+)
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TreePlan Student License, For Education Only TreePlan.com N RULE: P(D+ and T+) = P(D+|T+)* P(T+) ave P(D+ and T+) and P(T+) Solve for P(D+|T+) |T+)= P(D+ and T+) ÷ P(T+)
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TreePlan Student License, For Education Only TreePlan.com OBTAINED (GIVEN IN THIS CASE) Prior Conditional Probabilities 0 P(B | A) pulate. e outcome gray cells. d probabilities in the the table, Probability Trees, will autopopulate. screening ells.
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TreePlan Student License, For Education Only TreePlan.com 1 P(not B | A) 1 P(B | not A) 0 P(not B | not A) TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(A | B) 0.0000 P(not A | B) 1.0000 P(A | not B) #DIV/0! P(not A | not B) #DIV/0!
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TreePlan Student License, For Education Only TreePlan.com Event B B not B TOTALS B not B TOTALS Joint Probabilities P(A and B) 0 Event B: Test Result Example: Imagine th that 98% o negative re Next, assum If a person 1. what is t 2. what is t
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TreePlan Student License, For Education Only TreePlan.com P(A and not B) 0 P(not A and B) 1 P(not A and not B) 0 Joint Probabilities P(A and B) 0 P(not A and B) 1 P(A and not B) #DIV/0! P(not A and not B) #DIV/0! If a person 3. what is t 4. what is t Do these p
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TreePlan Student License, For Education Only TreePlan.com Ev A P(A and B) P(A and not B) P(A) Event A: Co A 0 0 0 CONTINGENCY TAB hat your company is thinking of implementi of the time, it shows a true positive result f esult for nonusers of the drug. me that based on past data, 0.5% of peop n selected at random tests positive for the the probability the person is a user of the the probability the person is not a user of
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TreePlan Student License, For Education Only TreePlan.com n selected at random tests negative for the the probability the person is not a user of the probability the person is a user of the probabilities make you concern that the es
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TreePlan Student License, For Education Only TreePlan.com vent A not A TOTALS P(not A and B) P(B) P(not A and not B) P(not B) P(not A) 1 ondition Status not A TOTALS 1 1 0 0 1 1 BLE ting drug testing. There is a drug test that is 98% accurate, meaning for someone using the drug, and 98% of the time, it shows a true ple within your company use the drug. drug,... drug? the drug? Diag abse typically the test which is
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TreePlan Student License, For Education Only TreePlan.com e drug,... the drug? drug? stimates for accuracy of the test are wrong?
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TreePlan Student License, For Education Only TreePlan.com g gnostic Testing: When we discuss diagnostic testing in medicine, our events are ence/presence of disease and positive or negative test result. Each company will y publish a sensitivity and specificity which are characteristics of the test and once is performed, patients are generally counseled based on the PPV and NPV of a test s based on the prevalence of the disease. The calculations are still the same as any other screening test, we have discussed thus far. Try the example below.
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TreePlan Student License, For Education Only TreePlan.com Name Value Prob Pred Kind NS TreePlan 0 0 0 E 2 0 E 2 0 E 2 1 T 0 1 T 0 2 T 0 2 T 0
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TreePlan Student License, For Education Only TreePlan.com S1 S2 S3 S4 S5 Row 1 2 0 0 0 9 3 4 0 0 0 4 5 6 0 0 0 14 0 0 0 0 0 2 0 0 0 0 0 7 0 0 0 0 0 12 0 0 0 0 0 17
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TreePlan Student License, For Education Only TreePlan.com Event Disease Disease (A) Using drug No Disease (not A) Not using drug Screening Test Test positive (B) Test positive Test negative (not B) Test negative Given (enter as pr P(Test positive | Using drug) 0.98 P(Test negative | Not using drug) 0.98 P(Using drug) 0.005 Solution Probability P(Using drug | Test positive) 19.758% P(Not using drug | Test positive) 80.242% P(Using drug | Test negative) 0.010% P(Not using drug | Test negative) 99.990% P(Test positive) 2.48% P(Test negative) 97.52% PRIOR PROBABIL Enter data into the gray boxes. The contingency table, probabi
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TreePlan Student License, For Education Only TreePlan.com Using drug Not using drug POSTERIOR PROB Test positive Test negative MULTIPLICATION RULE: P(T P(T+) = P
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TreePlan Student License, For Education Only TreePlan.com MULTIPLICATION We ha P(D+|
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TreePlan Student License, For Education Only TreePlan.com roportions) True Positive Rate=Sensitivity True Negative Rate=Specificity Prevalence ns Positive Predictive Value (PPV) Negative Predictive Value (NPV) Test Positive Rate Test Negative Rate LITIES AFTER THE NEW INFORMATION IS O Test positive P(Using drug) ility tree and conditional probabilities will autopop PROBABILITY TREES To use this Calculator Step 1: Change the labels for the to be predicted (Event A) in the g Step 3: Enter the required gray cells. The contingency and Solutions table w Step 2: Change the labels for the test (Event B) in the gray ce
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TreePlan Student License, For Education Only TreePlan.com 0.005 Test negative Test positive P(Not using drug) 0.995 Test negative BABILITIES AFTER THE NEW INFORMATION IS OBT Using drug P(Test positive) 0.0248 Not using drug Using drug P(Test negative) 0.9752 Not using drug T+|D+) * P(D+) = P(D+ and T+) = P(D+|T+)* P(T+) P(D+ and T+) + P(D- and T+)
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TreePlan Student License, For Education Only TreePlan.com N RULE: P(D+ and T+) = P(D+|T+)* P(T+) ave P(D+ and T+) and P(T+) Solve for P(D+|T+) |T+)= P(D+ and T+) ÷ P(T+)
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TreePlan Student License, For Education Only TreePlan.com OBTAINED (GIVEN IN THIS CASE) Prior Conditional Probabilities 0.98 P(Test positive | Using drug) pulate. e outcome gray cells. d probabilities in the the y table, Probability Trees, will autopopulate. screening ells.
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TreePlan Student License, For Education Only TreePlan.com 0.02 P(Test negative | Using drug) 0.02 P(Test positive | Not using drug) 0.98 P(Test negative | Not using drug) TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(Using drug | Test positive) 0.1976 P(Not using drug | Test positive) 0.8024 P(Using drug | Test negative) 0.00010 P(Not using drug | Test negative) 0.99990
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TreePlan Student License, For Education Only TreePlan.com
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TreePlan Student License, For Education Only TreePlan.com Event B Test positive Test negative TOTALS Test positive Test negative TOTALS Joint Probabilities P(Using drug and Test positive) 0.0049 Event B: Test Result Exampl Imagine that 98% negative Next, as If a pers 1. what 2. what If a pers 3. what
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TreePlan Student License, For Education Only TreePlan.com P(Using drug and Test negative) 0.0001 P(Not using drug and Test positive) 0.0199 P(Not using drug and Test negative) 0.9751 Joint Probabilities P(Using drug and Test positive) 0.0049 P(Not using drug and Test positive) 0.0199 P(Using drug and Test negative) 0.0001 P(Not using drug and Test negative) 0.9751 Solution If a per 1. the 2. the If a per 1. the 2. the Wow! T the dru (0.5%). tested. On the using th These p Specific conditio as the p become 3. what 4. what Do these
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TreePlan Student License, For Education Only TreePlan.com Ev Using drug P(Using drug and Test positive) P(Using drug and Test negative) P(Using drug) Event A: Co Using drug 0.0049 0.0001 0.005 CONTINGENCY TAB le: e that your company is thinking of impleme % of the time, it shows a true positive resu e result for nonusers of the drug. ssume that based on past data, 0.5% of pe son selected at random tests positive for t is the probability the person is a user of th is the probability the person is not a user son selected at random tests negative for t is the probability the person is not a user
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TreePlan Student License, For Education Only TreePlan.com n: rson selected at random tests positive probability that the person is actually probability that the person is not usin rson selected at random tests negative probability that the person is not usin probability that the person is using th That slso means that there is an 80% c ug despite the test having high accura . A test can be highly accurate but hav other hand, the NPV is very high mea he drug than it is at predicting that so probabilities do not indicate that the e city are inherent properties of the test on being tested. If the prevalence wer prevalence of drug use. In that case, t es 99.13% (compared to 99.99%). is the probability the person is not a user is the probability the person is a user of th e probabilities make you concern that the
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TreePlan Student License, For Education Only TreePlan.com vent A Not using drug TOTALS P(Not using drug and Test positive) P(Test positive) P(Not using drug and Test negative) P(Test negative) P(Not using drug) 1 ondition Status Not using drug TOTALS 0.0199 0.0248 0.9751 0.9752 0.995 1 BLE Diag abse typicall the test which i enting drug testing. There is a drug test that is 98% accurate, meani ult for someone using the drug, and 98% of the time, it shows a true eople within your company use the drug. the drug,... he drug? of the drug? the drug,... of the drug?
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TreePlan Student License, For Education Only TreePlan.com e for the drug using the drug is 19.76% (PPV) ng the drug is 80.24% e for the drug ng the drug is 99.99% (NPV) he drug is 0.01% chance that someone with a positive drug screen is not using acy. The main driver of this is the LOW prevalence of drug use ve a low PPV given the low prevalence of the condition being aning that the test is better at predicting that someone is not omeone is using the drug. estimates for the accuracy of the test are wrong. Sensitivity a t whereas NPV and PPV are influenced by the prevalence of th re higher, the PPV goes up and the NPV goes down. Enter 30% the PPV becomes 95.46% (compared to 19.76%) and the NPV of the drug? he drug? e estimates for accuracy of the test are wrong?
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TreePlan Student License, For Education Only TreePlan.com gnostic Testing: When we discuss diagnostic testing in medicine, our events are ence/presence of disease and positive or negative test result. Each company will ly publish a sensitivity and specificity which are characteristics of the test and once is performed, patients are generally counseled based on the PPV and NPV of a test is based on the prevalence of the disease. The calculations are still the same as any other screening test, we have discussed thus far. Try the example below. ing e
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TreePlan Student License, For Education Only TreePlan.com g ers and he % V
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TreePlan Student License, For Education Only TreePlan.com ID 0 1 2 3 4 5 6
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TreePlan Student License, For Education Only TreePlan.com Name Value Prob Pred Kind NS TreePlan 0 0 0 E 2 0 E 2 0 E 2 1 T 0 1 T 0 2 T 0 2 T 0
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TreePlan Student License, For Education Only TreePlan.com S1 S2 S3 S4 S5 Row 1 2 0 0 0 9 3 4 0 0 0 4 5 6 0 0 0 14 0 0 0 0 0 2 0 0 0 0 0 7 0 0 0 0 0 12 0 0 0 0 0 17
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TreePlan Student License, For Education Only TreePlan.com Col Mark 1 1 5 1 5 1 9 1 9 1 9 1 9 1
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