DDE-1/Statistics.md

@@ -44,7 +44,28 @@ p(E_2|E_1)= p(E_2);
 
 ### Bayesian statistics
 
-Probability of $`E_1`$ to occur, $`p(E_1)`$ can be represented as a weighted average of conditional probabilities: (i) probability of observing $`E_1`$ given that $`E_2`$ is observed, (ii) probability of observing $`E_1`$ given that $`E_2`$ is not observed.
+Probability of $`E_1`$ to occur, $`p(E_1)`$, can be represented as a weighted average of conditional probabilities: (i) probability of observing $`E_1`$ given that $`E_2`$ is observed, (ii) probability of observing $`E_1`$ given that $`E_2`$ is not observed:
+
+```math
+p(E_1) = p(E_1E_2) + p(E_1E_2^c) 
+```
+This is best seen over a [Venn diagram](https://en.wikipedia.org/wiki/Venn_diagram). You can see the spaces occupied by the event rules easily (c: complement)
+
+<div align="center">
+<img src="uploads/262d0b21f39beae9a935822bb21456b5/s1.png"  width="330">
+</div>
+
+If we replace the probabilities with conditional expressions discussed above, we get the following:
+
+```math
+p(E_1) = p(E_1|E_2)p(E_2) + p(E_1|E_2^c) p(E_2^c)
+```
+
+```math
+p(E_1) = p(E_1|E_2)` + p(E_1|E_2^c)(1-p(E_2))
+```
+
+This is exactly what we described in text at the beginning, where the weights are $`p(E_2)`$ and $`1-p(E_2)`$, respectively.
 
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