requires manipulating counterfactual information—not what the current known states are, but what are the next possible states.

Inside , there is already a few principles which are counterfactual.

1. Conservation of energy: a perpetual machine is *impossible
2. Second law: its impossible to convert all heat into useful work
3. Heisenberg’s uncertainty: its impossible to copy reliable all states of a qubit

With the impossibles, we can make the possible.

counting asks: “how many possible outcomes satisfy an event?” You create a “generative story” to think about how you can count for the total choices.

(like, 3 times will we roll even in a regular fair dice)

step rule of counting

• CS157 Introduction to Logic; Carta average 3.9, “Only take if you are actually interested in logic or this is an absolute requirement for your degree that you can’t dodge. Class structure is very tricky as your entire grade depends on how you do on 3 exams of 5 questions each.”

\begin{equation} cov(x,y) = E[(X-E[X])(Y-E[Y])] = E[XY]-E[X]E[Y] \end{equation}

(the derivation comes from FOIling the two terms and applying properties of expectation.

we want to consider: if a point goes way beyond its expectation, does the corresponding point change for another?

\begin{equation} (x-E[x])(y-E[y]) \end{equation}

if both points are varying .

Instead of using this unbounded value, we sometimes use a normalized value named correlation:

\begin{equation} \rho(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}} \end{equation}

we can express this in terms of a covariance matrix

• diagonal slots: how do the variables vary with themselves
• off-diagonal slots: how do the variables vary with each other