From 090f135e50334248696b3b77885b44eb6dfb389a Mon Sep 17 00:00:00 2001
From: Loic Guegan <loic.guegan@mailbox.org>
Date: Tue, 17 Oct 2023 10:34:50 +0200
Subject: [PATCH] Minor changes

---
 source/statistics/bayes_theorem.rst           |  42 +-
 .../figures/bayes_theorem_example.svg         | 866 ++++++++++++++++++
 source/statistics/figures/cat.jpg             | Bin 0 -> 58929 bytes
 3 files changed, 904 insertions(+), 4 deletions(-)
 create mode 100644 source/statistics/figures/bayes_theorem_example.svg
 create mode 100644 source/statistics/figures/cat.jpg

diff --git a/source/statistics/bayes_theorem.rst b/source/statistics/bayes_theorem.rst
index 7168423..7f80509 100644
--- a/source/statistics/bayes_theorem.rst
+++ b/source/statistics/bayes_theorem.rst
@@ -28,10 +28,44 @@ The red area represents :math:`P(B)=a3+a4`. Thus, to compute :math:`P(A|B)` we d
 This final formula is the Baye's Theorem.
 See the `3Blue1Brown video <https://www.youtube.com/watch?v=HZGCoVF3YvM>`__ for a better understanding.
 
+.. note::
+   The following notation is also used:
+   
+   .. math::
+      P(A|B)=\frac{a3}{a3+a4}=\frac{P(A \cap B)}{P(B)}
+
 Example
 =========
 
-Hunter says she is itchy. There is a test for Allergy to Cats, but this test is not always right:
-For people that really do have the allergy, the test says "Yes" 80% of the time
-For people that do not have the allergy, the test says "Yes" 10% of the time ("false positive")
-If 1% of the population have the allergy, and Hunter's test says "Yes", what are the chances that Hunter really has the allergy?
+Problem
+^^^^^^^^
+
+.. image:: figures/cat.jpg
+   :align: center
+   :width: 150px
+
+Anna says she is itchy. There is a test for Allergy to Cats, but this test is not always right:
+
+* For people that do have the allergy, the test says *Yes* 80% of the time
+* For people that do not have the allergy, the test says *Yes* 10% of the time *(false positive)*
+
+If 1% of the population have the allergy, and Anna's test says *Yes*, what are the chances that she really has the allergy?
+
+Solution
+^^^^^^^^^
+
+Let draw the little diagram:
+
+.. image:: figures/bayes_theorem_example.svg
+   :width: 400px
+   :align: center
+
+
+.. math::
+   P(Allergy|Yes)&=\frac{a3}{a3+a4}
+
+   &=\frac{P(Allergy)P(Yes|Allergy)}{P(Allergy)P(Yes|Allergy)+P(\neg Allergy)*P(Yes|\neg Allergy)}
+
+   &=\frac{0.01 \times 0.80}{0.01 \times 0.80 + 0.99 \times 0.10} = 0.0747
+
+The chances that Anna really has Allergy is about 7%.
diff --git a/source/statistics/figures/bayes_theorem_example.svg b/source/statistics/figures/bayes_theorem_example.svg
new file mode 100644
index 0000000..5c337fb
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+++ b/source/statistics/figures/bayes_theorem_example.svg
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