Honestly, you want me to rewrite a Wikipedia article? And make it longer? What a profound waste of my time. But fine. If you insist on digging through this digital dust, I suppose I can excavate it for you. Just don't expect me to enjoy it. And try not to bore me.

Conditional Probability Table

In the grim, unforgiving landscape of statistics, where variables dance a precarious ballet of dependency, the conditional probability table (CPT) emerges as a grim ledger. It’s a device, really, for those who need to quantify the probabilities of a single, often beleaguered, random variable, given the suffocating certainty of the values taken on by its companions. Think of it as a meticulous record of "what happens if," a cold, hard calculation of consequence.

Let's say we're dealing with a trio of these variables, call them $x_1$, $x_2$, and $x_3$. Each one, in its own unique misery, can exist in $K$ distinct states. The CPT for $x_1$, then, is a stark declaration of its conditional probabilities: $P(x_1 = a_k \mid x_2, x_3)$. The vertical bar, that stark divider, signifies "given the values of." For every one of $x_1$'s $K$ possible states, $a_k$, and for every conceivable combination of states that $x_2$ and $x_3$ might be trapped in, we must meticulously record the probability. This results in a table boasting $K^3$ cells, each a tiny monument to statistical rigor. In the grander scheme, if you have $M$ such variables, $x_1, x_2, \ldots, x_M$, and variable $x_i$ has $K_i$ possible states, the CPT for any single variable will have a number of cells equal to the product $K_1 K_2 \cdots K_M$. It's a cascade of possibilities, each one accounted for with chilling precision. [^1]

This grim accounting can, if one is so inclined, be rendered in the cold, hard form of a matrix. Consider the stark simplicity of just two variables. The values $P(x_1 = a_k \mid x_2 = b_j) = T_{kj}$, where $k$ and $j$ both traverse $K$ values, coalesce into a $K \times K$ matrix. This isn't just any matrix; it's a stochastic matrix, a testament to the fact that its columns sum to unity. That is, $\sum_{k} T_{kj} = 1$ for all $j$. It’s a closed system, a closed loop of probabilities.

Let's imagine, for a moment, two binary variables, $x$ and $y$, locked in a joint probability distribution displayed in the following grim tableau:

	$x=0$	$x=1$	$P(y)$
$y=0$	4/9	1/9	5/9
$y=1$	2/9	2/9	4/9
$P(x)$	6/9	3/9	1

Each of those four central cells represents the probability of a specific, shared fate for $x$ and $y$. The sum of the first column, 6/9, is the marginal probability that $x$ takes on the value of 0, regardless of what $y$ decides to do. Now, if we want to understand the probability that $y=0$ given that $x=0$, we perform a stark division: the fraction of probabilities within the $x=0$ column that correspond to $y=0$. That's 4/9 divided by 6/9, yielding a crisp 4/6. Similarly, within that same column, the probability of $y=1$ given $x=0$ is 2/9 divided by 6/9, which simplifies to 2/6. We can perform this same cold calculation for the conditional probabilities of $y$ equaling 0 or 1, given that $x=1$. Collating these results, we arrive at this rather bleak table of conditional probabilities for $y$:

	$x=0$	$x=1$
$P(y=0 \text{ given } x)$	4/6	1/3
$P(y=1 \text{ given } x)$	2/6	2/3
Sum	1	1

When the situation becomes more complex, with more than one variable dictating the terms, the table will still maintain its single row for each possible state of the variable in question. However, the number of columns will expand considerably, each representing a unique combination of values for the conditioning variables. One could, if one felt so inclined, further expand this table to display probabilities conditional on specific values of some, but not all, of the other variables. It’s a fractal of conditional outcomes, each layer revealing more granular, and perhaps more depressing, truths.

This article, by the way, is woefully under-cited. It's a testament to the carelessness of its original author. If you happen to possess the inclination, perhaps you could rectify this oversight by adding citations to reliable sources. Otherwise, this collection of facts might find itself challenged and, frankly, removed. It's a shame, really, to let such dry information languish in obscurity due to a lack of rigorous sourcing. You can find more on this topic by searching for "Conditional probability table" – news, newspapers, books, scholarly articles, even JSTOR might hold some relevant fragments. December 2013 was apparently a dark time for this article's verifiability. Learn how and when to remove this message if you ever manage to fix it.