The document discusses three hypothetical examples of using Bayes' rule to calculate the probability of a condition being true given a positive test result. In the first two examples, which deal with Edwards syndrome and Down syndrome, it is shown that despite the tests having high sensitivity and specificity, the actual probability of the condition being present is quite low, around 1.7% and 9% respectively. In the third example dealing with blue eyes, the high base rate in the population results in a 95.2% chance of blue eyes given a positive test.

1. Hypothetical Example: 1
• Non-Invasive Prenatal Testing (NIPT) is an increasingly
popular technique that screens for chromosomal
abnormalities during pregnancy.
• Huge benefits with few risks.
• They are highly accurate, with 99% of abnormalities
identified by the test.
• Your (or your partner’s) test comes back saying your baby
has Edwards syndrome (trisomy 18)…
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2. Hypothetical Example: 1
• What is the probability that your baby actually has
Edwards syndrome?
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3. Hypothetical Example: 1
• What is the probability that your baby actually has
Edwards syndrome?
• Suppose the sensitivity of the test is high:
• 99% of fetuses with trisomy are detected.
• Suppose the specificity of the test is also high:
• 99% of fetuses with no trisomies test negative.
• Given your positive test and this information, does
your fetus have a high or low probability of actually
having a chromosomal abnormality?
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4. 3
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Hypothetical Example: 1

5. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
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6. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify…
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7. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify…
• Recall:
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8. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify…
• Recall:
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9. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify…
• Recall:
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10. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify…
• Recall:
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11. Hypothetical Example: 1
• Bayes Rule comes to the rescue!!
• Let A be the event “Have trisomy 18”
• Let B be the event “Test Positive for trisomy 18”
• We want P(A | B) in terms we can easily quantify:
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“Probability of
having trisomy 18
given positive test”
“Probability of having
positive test given
you have trisomy 18”
“Probability
of having
trisomy 18”
“Probability of
testing positive
for trisomy 18”

12. Hypothetical Example: 1
• A = “Have trisomy 18”; B = “Test Positive for trisomy 18”
• In the USA, P(A) = Pr(have trisomy 18) = 1/6000 = 1.7e-4
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
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13. Hypothetical Example: 1
• A = “Have trisomy 18”; B = “Test Positive for trisomy 18”
• In the USA, P(A) = Pr(have trisomy 18) = 1/6000 = 1.7e-4
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
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14. Hypothetical Example: 1
• A = “Have trisomy 18”; B = “Test Positive for trisomy 18”
• In the USA, P(A) = Pr(have trisomy 18) = 1/6000 = 1.7e-4
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
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15. Hypothetical Example: 1
• A = “Have trisomy 18”; B = “Test Positive for trisomy 18”
• In the USA, P(A) = Pr(have trisomy 18) = 1/6000 = 1.7e-4
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
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16. Hypothetical Example: 1
• A = “Have trisomy 18”; B = “Test Positive for trisomy 18”
• In the USA, P(A) = Pr(have trisomy 18) = 1/6000 = 1.7e-4
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
• Thus, there is only a ~1.7% chance your fetus has
trisomy 18, despite the high sensitivity and specificity!!
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17. Hypothetical Example: 2
• Down Syndrome: trisomy 21.
• A = “Have trisomy 21”; B = “Test Positive for trisomy
21”
• In the USA, P(A) = Pr(have trisomy 21) = 1/1000 = 1e-3
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18. Hypothetical Example: 2
• Down Syndrome: trisomy 21.
• A = “Have trisomy 21”; B = “Test Positive for trisomy
21”
• In the USA, P(A) = Pr(have trisomy 21) = 1/1000 = 1e-3
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
• Thus, there is only a 9% chance your fetus has trisomy
21, despite the high sensitivity and specificity!!
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19. Hypothetical Example: 3
• A = “Have blue eyes”; B = “Test Positive for OCA2
variant”
• In the USA, P(A) = Pr(have blue eyes) = 1/6 = 0.166
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
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20. Hypothetical Example: 3
• A = “Have blue eyes”; B = “Test Positive for OCA2
variant”
• In the USA, P(A) = Pr(have blue eyes) = 1/6 = 0.166
• Sensitivity: P(B|A) = 0.99; Specificity: P(BC|AC)=0.99
• Thus, there is a 95.2% chance your baby will have blue
eyes!
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