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@ -28,10 +28,44 @@ The red area represents :math:`P(B)=a3+a4`. Thus, to compute :math:`P(A|B)` we d
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This final formula is the Baye's Theorem.
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See the `3Blue1Brown video <https://www.youtube.com/watch?v=HZGCoVF3YvM>`__ for a better understanding.
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.. note::
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The following notation is also used:
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.. math::
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P(A|B)=\frac{a3}{a3+a4}=\frac{P(A \cap B)}{P(B)}
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Example
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=========
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Hunter says she is itchy. There is a test for Allergy to Cats, but this test is not always right:
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For people that really do have the allergy, the test says "Yes" 80% of the time
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For people that do not have the allergy, the test says "Yes" 10% of the time ("false positive")
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If 1% of the population have the allergy, and Hunter's test says "Yes", what are the chances that Hunter really has the allergy?
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Problem
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^^^^^^^^
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.. image:: figures/cat.jpg
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:align: center
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:width: 150px
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Anna says she is itchy. There is a test for Allergy to Cats, but this test is not always right:
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* For people that do have the allergy, the test says *Yes* 80% of the time
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* For people that do not have the allergy, the test says *Yes* 10% of the time *(false positive)*
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If 1% of the population have the allergy, and Anna's test says *Yes*, what are the chances that she really has the allergy?
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Solution
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^^^^^^^^^
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Let draw the little diagram:
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.. image:: figures/bayes_theorem_example.svg
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:width: 400px
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:align: center
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.. math::
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P(Allergy|Yes)&=\frac{a3}{a3+a4}
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&=\frac{P(Allergy)P(Yes|Allergy)}{P(Allergy)P(Yes|Allergy)+P(\neg Allergy)*P(Yes|\neg Allergy)}
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&=\frac{0.01 \times 0.80}{0.01 \times 0.80 + 0.99 \times 0.10} = 0.0747
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The chances that Anna really has Allergy is about 7%.
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866
source/statistics/figures/bayes_theorem_example.svg
Normal file
866
source/statistics/figures/bayes_theorem_example.svg
Normal file
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@ -0,0 +1,866 @@
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