Bayes' theorem is very useful for diagnosis and for accountability. It's easy to calculate, difficult to understand because of not intuitive. This article is using concept of odds with figures to study Bayes' theorem easily.

1. Bayes' Theorem
Easy to Understand
with odds and figures
Misaki Yositake
M.Ed. Mathematics
Bayes' theorem is very useful for diagnosis and for accountability. It's easy to calculate,
difficult to understand because of not intuitive. This article is using concept of odds with
figures to study Bayes' theorem easily.
Definition 1
Figure 1
Definitions ( * )
Sensitivity— The proportion of people with the disease who are correctly identified by a
positive test result (“true positive rate”)
Specificity— The proportion of people free of the disease who are correctly identified by a
negative test result (“true negative rate”)
Pretest probability (prevalence)—The probability that an individual has the target disorder
before the test is carried out
Post-test probability—The probability that an individual with a specific test result has the
target condition (post-test odds/[1+post-test odds]) or
Pretest odds—The odds that an individual has the target disease before the test is carried
out (pretest probability/[1-pretest probability])
Post-test odds—The odds that a patient has the target disease after being tested .
Positive predictive value (PPV)—The proportion of individuals with positive test results who
have the target condition. This equals the post-test probability given a positive test result
Negative predictive value (NPV)—The proportion of individuals with negative test results who
do not have the target condition. This equals one minus the post-test probability given a
negative test result.
For example, the probability that toss a dice is 6 is one-sixth. Odds ratio is 1:5. o is favor,
and x is not favor.
o xxxxx
'Odds' are an expression of probabilities. For example, the probability that a random day
is a Sunday is one-seventh (1/7). Odds ratio is 6 to 1, 6-1, 6:1, or 6/1.

2. Now assumed pre test probability is 0.003. (Prevalence = 0.003)
Number of targeting people is 1000.
So, positive pretest odds is 3, pretest negative odds is neary 950.
Figure 2
Assuming sensitivity and specificity is 0.95. After positive test, we will get number of
true positive and number of false positive. Sensitivity is 0.95, we get that true positive
odds is 3 approximately. Specificity 0.95, so false positive is 1 - 0.95=0.05.
True odds is 3*0.95 = 3, false odds is 1000 * 0.05 = 50.
Figure 3
Finally we get post-test-odds ratio. It is 3 : 50. Positive predictive value is 3/ 50 =0.6 .
Conclusion
It's very intuitive aren't you?
Reference( * ) "Ruling diagnoses in and out with SpPIns and SnNOuts"
1 Very small number of event(people with disease) induces very small number true positive
post test odds number. It is p.
2. Very large event(healthy people) induces large false positive odds number relatively. It is q.
3 Therefor, we get p < q.
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True post test odds = 3
ooo
False post test odds = 50
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3. M Egger, Department of Social and Preventive Medicine, University of Bern,
Finkenhubelweg 11, CH-3012 Berne, Switzerland
http://www.bmj.com/content/329/7459/209.full