Extra Bayes Theorem Problems (Solutions)

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Virginia Commonwealth University *

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Biology

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Nov 24, 2024

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TreePlan Student License, For Education Only TreePlan.com Event Disease Disease (A) Genetic Defect No Disease (not A) No Genetic Def Screening Test Test positive (B) Test positive Test negative (not B) Test negative Given (enter as pr P(Test positive | Genetic Defect) 0.9 P(Test negative | No Genetic Defect) 0.904 P(Genetic Defect) 0.01 Solution Probability P(Genetic Defect | Test positive) 8.651% P(No Genetic Defect | Test positive) 91.349% P(Genetic Defect | Test negative) 0.112% P(No Genetic Defect | Test negative) 99.888% P(Test positive) 10.40% P(Test negative) 89.60% PRIOR PROBABIL P( Enter data into the gray boxes. The contingency
TreePlan Student License, For Education Only TreePlan.com Genetic Defect No Genetic Defect POSTERIOR PROB Test positive Test negative 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 t fect 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 (Genetic Defect) y table, probability tree and conditional probabilitie 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 0.01 Test negative Test positive P(No Genetic Defect) 0.99 Test negative BABILITIES AFTER THE NEW INFORMATION IS OBT Genetic Defect P(Test positive) 0.10404 No Genetic Defect Genetic Defect P(Test negative) 0.89596 No Genetic Defect 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 0.9 P(Test positive | Genetic Defect) es will autopopulate. 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 0.1 P(Test negative | Genetic Defect) 0.096 P(Test positive | No Genetic Defect) 0.904 P(Test negative | No Genetic Defect) TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(Genetic Defect | Test positive) 0.0865 P(No Genetic Defect | Test positive) 0.9135 P(Genetic Defect | Test negative) 0.00112 P(No Genetic Defect | Test negative) 0.99888
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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(Genetic Defect and Test positive) 0.009 Event B: Test Result
TreePlan Student License, For Education Only TreePlan.com P(Genetic Defect and Test negative) 0.001 P(No Genetic Defect and Test positive) 0.09504 P(No Genetic Defect and Test negative) 0.89496 Joint Probabilities P(Genetic Defect and Test positive) 0.009 P(No Genetic Defect and Test positive) 0.09504 P(Genetic Defect and Test negative) 0.001 P(No Genetic Defect and Test negative) 0.89496
TreePlan Student License, For Education Only TreePlan.com Ev Genetic Defect P(Genetic Defect and Test positive) P(Genetic Defect and Test negative) P(Genetic Defect) Event A: Co Genetic Defect 0.009 0.001 0.01 0.14 CONTINGENCY TAB Example: Your company is testing a new test t trials to date: •1% of people have a certain genetic •90% of tests for the gene detect th •9.6% of the tests are false positives If a person gets a positive test result Solution: Information Given in Probability l P(Genetic Defect)=P(A)=0.01 [Pr
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TreePlan Student License, For Education Only TreePlan.com P(Genetic Defect)=P(A)=0.01 [Pr P(Test positive|Genetic Defect)= P(Test positive)|No Genetiic Defe Calculations: 1. 1st need P(Test negative|No G P(Test negative|No Genetic 2. P(Genetic Defect|Test positive
TreePlan Student License, For Education Only TreePlan.com vent A No Genetic Defect TOTALS P(No Genetic Defect and Test positive) P(Test positive) P(No Genetic Defect and Test negative) P(Test negative) P(No Genetic Defect) 1 ondition Status No Genetic Defect TOTALS 0.09504 0.10404 0.89496 0.89596 0.99 1 BLE to predict a genetic defect. The following data is available from your tic defect. he defect (true positives). s. t, what is the probability they actually have the genetic defect? language: revalence] Diag abse typicall the test which i
TreePlan Student License, For Education Only TreePlan.com revalence] =P(B|A)= 0.90 ect)=P(B|not A)=0.096 Genetic Defect) to enter into the calculator. c Defect)= P(not B|not A)=1-0.096=0.904. This is the specific e)=P(A|B)= 8.65%
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TreePlan Student License, For Education Only TreePlan.com r 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.
TreePlan Student License, For Education Only TreePlan.com city.
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) Breast Cancer No Disease (not A) No Breast Canc Screening Test Test positive (B) Positive Mamm Test negative (not B) Negative Mamm Given (enter as pr P(Positive Mammogram | Breast Cancer) 0.9 P(Negative Mammogram | No Breast Can 0.92 P(Breast Cancer) 0.01 Solution Probability P(Breast Cancer | Positive Mammogram) 10.204% P(No Breast Cancer | Positive Mammogra 89.796% P(Breast Cancer | Negative Mammogram) 0.110% P(No Breast Cancer | Negative Mammogr 99.890% P(Positive Mammogram) 8.82% P(Negative Mammogram) 91.18% PRIOR PROBABIL P Enter data into the gray boxes. The contingency ta
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TreePlan Student License, For Education Only TreePlan.com Breast Cancer No Breast Cancer POSTERIOR PROB Positive Mammogram Negative Mammogram 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 cer mogram mogram 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 Positive Mammogram P(Breast Cancer) able, probability tree and conditional probabilities w 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 0.01 Negative Mammogram Positive Mammogram P(No Breast Cancer) 0.99 Negative Mammogram BABILITIES AFTER THE NEW INFORMATION IS OBT Breast Cancer P(Positive Mammogram) 0.0882 No Breast Cancer Breast Cancer P(Negative Mammogram) 0.9118 No Breast Cancer 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 0.9 P(Positive Mammogram | Breast Cancer) will autopopulate. 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 0.1 P(Negative Mammogram | Breast Cancer P 0.08 P(Positive Mammogram | No Breast Can P 0.92 P(Negative Mammogram | No Breast Can P( TAINED (CALCULATED IN THIS CASE) Posterior Conditional Probabilities P(Breast Cancer | Positive Mammogram) 0.1020 P(No Breast Cancer | Positive Mammogram) P 0.8980 P(Breast Cancer | Negative Mammogram) P 0.00110 P(No Breast Cancer | Negative Mammogram) P( 0.99890
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TreePlan Student License, For Education Only TreePlan.com Event B Positive Mammogram Negative Mammogram TOTALS Positive Mammogram Negative Mammogram TOTALS Joint Probabilities P(Breast Cancer and Positive Mammogram) 0.009 Event B: Test Result
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TreePlan Student License, For Education Only TreePlan.com P(Breast Cancer and Negative Mammogram) 0.001 P(No Breast Cancer and Positive Mammogram) 0.0792 (No Breast Cancer and Negative Mammogram) 0.9108 Joint Probabilities P(Breast Cancer and Positive Mammogram) 0.009 P(No Breast Cancer and Positive Mammogram) 0.0792 P(Breast Cancer and Negative Mammogram) 0.001 (No Breast Cancer and Negative Mammogram) 0.9108
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TreePlan Student License, For Education Only TreePlan.com Ev Breast Cancer P(Breast Cancer and Positive Mammogram) P(Breast Cancer and Negative Mammogram) P(Breast Cancer) Event A: Co Breast Cancer 0.009 0.001 0.01 CONTINGENCY TAB Example: Your company is marketing a mamm probability that a woman over age •One percent of women over 50 hav •Ninety percent of women who hav •Eight percent of women will have f Solution: The following information that is P(Breast cancer)=P(A)=0.01 [Pre
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TreePlan Student License, For Education Only TreePlan.com P(Breast cancer)=P(A)=0.01 [Pre P(Positive Mammogram|Breast Ca P(Positive Mamogram|No Breast C Calculations: 1. 1st need P(Negative Mammogr P(Negative Mammogram|N 2. P(Breast Cancer|Positive Mamm
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TreePlan Student License, For Education Only TreePlan.com vent A No Breast Cancer TOTALS P(No Breast Cancer and Positive Mammogram) P(Positive Mammogram P(No Breast Cancer and Negative Mammogram) P(Negative Mammogra P(No Breast Cancer) 1 ondition Status No Breast Cancer TOTALS 0.0792 0.0882 0.9108 0.9118 0.99 1 BLE mogram machine. Given the following statistics, what is the 50 has cancer if she has a positive mammogram result? ve breast cancer. ve breast cancer test positive on mammograms. false positives given applies to women over age 50: evalence] Diag abse typicall the test which i
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TreePlan Student License, For Education Only TreePlan.com evalence] ancer)=P(B|A)= 0.90 Cancer)=P(B|not A)=0.08 ram|No Breast Cancer) to enter into the calculator. No Breast Cancer)= P(not B|not A)=1-0.08=0.92. This is the sp mogram)=P(A|B)= 10.20%
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TreePlan Student License, For Education Only TreePlan.com m) am) 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 pecificity.
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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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