feat: add a stat example to verify impl vs theoretical

DEV.md

@@ -87,4 +87,43 @@ Now it's similar to xxhash where you pass in a void pointer and a length to it.
 
 ## Fixing clangd
 
-Post the split, I needed to fix clangd as it did not
+Post the split, I needed to fix clangd as it did not register variables across files as I did not
+have a valid cmake. So added that and ran `cmake -S . -B build` and all worked fine.
+
+---
+
+# The Math
+
+Better to quickly go over the math as well
+n - no of keys added ( expected ), m -> filter size in bits, k -> no of hash functions
+
+Now there are m bits, and consider one insertion, and just one hash function
+Proba that it's still 0 is (1-1/m) as 1/m is the chance that bit is occupied and hence 1.
+Now there are n keys and k such hash functions. This takes you to
+
+(1-1/m) ^ nk
+
+and then P(some particular bit is set) is 1-(1-1/m)^nk
+
+For a test k, it's false positive if all of the k bits are set which gives you
+
+(1 - (1-1/m)^nk ) ^ k
+
+For large m, (1-1/x)^x ~= 1/e
+so the inner part becomes 1 - e^(-kn/m)
+
+This last step is incorrect as the assuming some bitstates to be independent
+P(bit A is 1 and bit B is 1) = P(bit A is 1) * P(bit B is 1)
+
+Now this looks correct except that it is not
+https://www.math.umd.edu/~immortal/CMSC420/notes/bloomfilters.pdf
+
+But that's besides the point for now.
+
+## My error probab function?
+
+Based off of Guava, it uses the actual free bits at runtime to give a "live" estimate
+bits filled fraction is P(any arbitrary bit is 1)
+False positive = all of those bits end up being 1 for some hash = P^k
+
+This has the same independence issue in assumption though.