Capstok — learn by doing

Why this matters

Conditional probability is the formalism for 'given that I now know $B$ , what is my probability of $A$ ?' Almost every applied use of probability in the wild — medical tests, spam filters, A/B test interpretation, fraud detection, Bayesian inference — is conditional. The reason this is one of the most-flubbed topics is that natural language conditioning is sloppy: 'the probability the patient has cancer given a positive test' is not the same number as 'the probability of a positive test given cancer,' and conflating those two is the engine that drives base-rate fallacies, prosecutor's fallacies, and bad PMs misreading dashboards. Internalize the formula now and these confusions evaporate.

Demo

Given two events $A, B$ with $P(B) > 0$ , the conditional probability of $A$ given $B$ is the proportion of $B$ 's probability mass that also falls inside $A$ — that is, you renormalize the universe to $B$ .

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Try it yourself

Run the snippet and verify $P(A) = 5/36$ , $P(B) = 1/6$ , $P(A \mid B) = 1/6$ . Read the last number aloud: 'given die-1 is 3, the chance the sum is 8 equals the chance die-2 is 5.' Sanity passes.

Prompt your AI

Use these three in order. Each builds on the one before.

1. Basics & terminology

In one paragraph, explain conditional probability $P(A \mid B)$ in plain words, using a medical-test example. Be explicit about the difference between $P(\text{disease} \mid \text{positive test})$ and $P(\text{positive test} \mid \text{disease})$.

2. Why it works (the mechanism)

Walk me step by step through *why* the conditional formula $P(A \mid B) = P(A \cap B) / P(B)$ has the form it does. Start from 'restrict the sample space to $B$ and renormalize so $P(B \mid B) = 1$.'

3. Advanced — application & what's next

When does $P(A \mid B) = P(A \mid B^c)$ hold? Show that this is equivalent to $A$ and $B$ being independent, and explain why this is a stronger statement than 'knowing $B$ does not change my belief about $A$.'

References

// main.go
package main

import "fmt"

func main() {
	// Build the sample space: all pairs (d1, d2) with d1, d2 in 1..6
	type roll [2]int
	var omega []roll
	for d1 := 1; d1 <= 6; d1++ {
		for d2 := 1; d2 <= 6; d2++ {
			omega = append(omega, roll{d1, d2})
		}
	}

	var A, B, AB []roll
	for _, w := range omega {
		if w[0]+w[1] == 8 {
			A = append(A, w) // event A: sum is 8
		}
		if w[0] == 3 {
			B = append(B, w) // event B: first die is 3
		}
		if w[0]+w[1] == 8 && w[0] == 3 {
			AB = append(AB, w)
		}
	}

	pA := float64(len(A)) / 36
	pB := float64(len(B)) / 36
	pAB := float64(len(AB)) / 36

	fmt.Printf("P(A) = %.4f\n", pA)
	fmt.Printf("P(B) = %.4f\n", pB)
	fmt.Printf("P(A|B) = %.4f\n", pAB/pB) // 1/6 — given die1=3, die2 must be 5
}

Run: go run main.go