DDE-1/Statistics.md

@@ -65,11 +65,52 @@ p(E_1) = p(E_1|E_2)p(E_2) + p(E_1|E_2^c) p(E_2^c)
 p(E_1) = p(E_1|E_2)p(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.
+This is exactly what we described in text at the beginning, where the weights are now explicitly seen as the $`p(E_2)`$ and $`1-p(E_2)`$, respectively. In other words, each conditional probability is scaled with respect to the probability of it is happening. 
+
+How is this useful to us? Let's discuss it over an example. We would like to know  the probability a person has a disease (e.g. Covid) given that the test result is positive (e.g. quick antigen tests). Assume that the test is not that accurate: 85% times gives true positives, while gives 3% false positives on healthy people. We suspect that 1% of the population has the disease in average. Question is, what is the probability of the getting a positive test results, if we are actually sick?  
+
+In order to simplify things, lets denote being tested positive and negative with + and - signs, where the sick and healthy people are symbolized as D and H, respectively. So, what we want to know:
+
+```math
+p(D|+) = p(D+)/p(+)
+```
+Note that:
+
+
+```math
+p(D|+) = p(+|D)p(D)
+```
+
+So, $`p(D|+)`$ becomes:
+
+```math
+p(D|+) = p(+|D)p(D)/p(+)
+```
+
+We now know how to evaluate $`p(+)`$:
+
+```math
+p(+|D) = p(+|D)p(D)/(p(+|D)p(D)+p(+|D^c)p(D^c))
+```
+
+If we put the numbers:
+
+
+```math
+p(+|D) = (0.85)(0.01)/((0.85)(0.01)+(0.03)(0.99))
+```
+
+```math
+p(+|D) = (0.0085)/(0.0382) = 0.22
+```
+
+It seems even if the test seems to be relatively accurate to capture sick people, the probability of being measured as a sick person given that you are sick is 22 %. There is also a very nice [visual illustration here]( https://seeing-theory.brown.edu/bayesian-inference/index.html#section1) that you can play with.
+
+How can we generalize the approach for N number of events?
 
 ...
 
-Bayes’s formula is utilized under the hood of several models we will learn throughout the lecture. Herein, we will update our understanding of the probabilistic world (i.e., a probability distribution) by making new observations. Let’s look at an example, for which we have [a visual illustration]( https://seeing-theory.brown.edu/bayesian-inference/index.html#section1) you can play with.
+Bayes’s formula is utilized under the hood of several models we will learn throughout the lecture.  Let’s look at an example, for which we have [a visual illustration]( https://seeing-theory.brown.edu/bayesian-inference/index.html#section1) you can play with.
 
 ### What is likelihood?