more notes

R/ch2.qmd

@@ -114,4 +114,39 @@ prop_of_excl_within_type |>
 
 The table above also shows the likelihoods for the case 
 when an article does not contain exclamation point in 
-the title.
+the title as well. It's really important to note that these are likelihoods,
+and its not the case that $L(B|A) + L(B^c|A) = 1$ as a matter of fact this
+value evaluates to a number less than one. However, since we have that
+$L(B|A) = .267$ and $L(B^c|A) = .022$ then we have gained additional 
+knowledge in knowing the use of "!" in a title is more compatible 
+with a fake news article than a real one. 
+
+Up to this point we can summarize our framework as follows
+
+| event      | $B$ | $B^c$ | Total |
+|-------     |-----|-------|------|
+| prior      | .4  |  .6   | 1    |
+| likelihood |.267 | .022  | .289 |
+
+Our next goal is come up with normalizing factors in order to build our 
+probability table:
+
+|      | $B$| $B^c$| Total |
+|------|----|------|-------|
+|$A$   | (1)| (2)  |       |
+|$A^c$ | (3)|  (4) |       |
+|Total | .4 | .6   | 1     |
+
+A couple things to note about our table (1) + (3) = .4 and (2) + (4) = .6.
+(1) + (2) + (3) + (4) = 1. 
+
+(1) $P(A \cap B) = P(A|B)P(B)$ we know the likelihood of $L(B|A) = P(A|B)$ and we also 
+know the prior so we insert these to get 
+$$ P(A \cap B) = P(A|B)P(B)  = .267 \times .4 = .1068$$
+
+(3) $P(A^c \cap B) = P(A^c|B)P(B)$ in this case we do know the prior $P(B) = .4$, but we 
+don't directly know the value of $P(A^c|B)$, however, we note that $P(A|B) + P(A^c|B) = 1$, 
+therefore we compute $P(A^c|B) = 1 - P(A|B) = 1 - .267 = .733$
+$$ P(A^c \cap B) = P(A^c|B)P(B)  = .733 \times .4 = .2932$$
+
+we now can confirm that $.1068 + .2932 = .4$