[Verse 1]
When two events might both occur
But some overlap you can't ignore
Union means A or B could be
Add them up but carefully
The intersection counts twice through
So subtract that overlap too
[Chorus]
Probability rules, they govern chance
Union adds minus the dance
Conditional narrows the space
Multiplication finds the place
Total probability breaks it down
These are the laws that run this town
[Verse 2]
Given B has come to pass
Now A's chances we must ask
Take their intersection neat
Divide by B's probability beat
The universe has gotten small
Only B events matter at all
[Chorus]
Probability rules, they govern chance
Union adds minus the dance
Conditional narrows the space
Multiplication finds the place
Total probability breaks it down
These are the laws that run this town
[Bridge]
When both events must coincide
Conditional times the other side
A given B times probability B
Or flip the script symmetrically
B given A times A alone
Same result, different zone
[Verse 3]
Total probability's the key
When paths branch out like a tree
Sum up A through each route
Conditional times absolute
Every possible way it flows
Add them up, that's how it goes
[Final Chorus]
Probability rules, they govern chance
Union adds minus the dance
Conditional narrows the space
Multiplication finds the place
Total probability breaks it down
Grammar of uncertainty's crown
[Outro]
From courtrooms to machines that learn
These axioms at every turn
The language of what might be
Mathematical certainty