Independent events — multiplication rule: Visual walkthrough

Before we start, three words we will draw, never assume:

Everything below is just those three pictures interacting.

Step 1 — Probability is area on a unit square

WHAT. Draw a square exactly $1$ wide and $1$ tall. Its total area is $1 \times 1 = 1$ . That " $1$ " is the total probability of everything — some outcome must happen.

WHY this picture. Probability is a share of certainty. Certainty $= 1$ . A square of area $1$ lets us represent any probability as a piece of that area — a big piece means "likely", a sliver means "rare". Area is the perfect tool because it adds and multiplies just like probabilities do.

PICTURE. The whole blueprint square is the sample space. We will slice it.

Figure — Independent events — multiplication rule

Step 2 — One event = a vertical strip

WHAT. Let event $A$ have probability $P (A)$ . Slice the square with a vertical line so the left strip has width $P (A)$ . Being inside that strip means " $A$ happened".

WHY. We chose width (the horizontal direction) for $A$ so that later we can give the height (vertical direction) to a second event $B$ . Two independent directions for two independent events — that separation is the entire secret, and we are setting it up now.

PICTURE. The amber strip's width is $P (A)$ ; its area is $P (A) \times 1 = P (A)$ . The right part (width $1 - P (A)$ ) is " $A$ did not happen" — the complement.

Step 3 — Conditional probability = re-measuring inside the strip

WHAT. Now bring in $B$ . Suppose we are told " $A$ already happened." That means we are standing inside the amber strip — the rest of the square is irrelevant. Within that strip, what fraction (of the strip's height) makes $B$ true? That fraction is $P (B ∣ A)$ .

WHY this tool. We need a way to talk about $B$ after $A$ is known. The honest definition is Conditional Probability: $P (B ∣ A) = \frac{P ( A \cap B )}{P ( A )} .$ Read the picture: it says "of the strip's area $P (A)$ , what share is also in $B$ ?" That share is exactly $P (A \cap B)$ sitting on top of $P (A)$ .

PICTURE. Inside the amber strip we shade the bottom portion cyan — its height (as a fraction of the strip) is $P (B ∣ A)$ . That cyan block is the overlap $A \cap B$ .

Step 4 — Rearrange: the General Multiplication Rule (still no assumptions)

WHAT. Multiply both sides of Step 3 by $P (A)$ . The $P (A)$ cancels on the right: $P (A \cap B) = P (B ∣ A) P (A) .$

WHY. We just want the area of the cyan block directly. Area of a rectangle $=$ width $\times$ height. Its width is $P (A)$ (the strip), its height is $P (B ∣ A)$ (the fraction of the strip that is cyan). Width times height — that's all this line says. It is true for any two events; nothing about independence yet. This is the General Multiplication Rule.

PICTURE. The cyan rectangle, labelled width $P (A)$ , height $P (B ∣ A)$ , area $=$ their product.

Step 5 — Impose independence: the height stops caring about $A$

WHAT. Independence is the special case where knowing $A$ does not change $B$ 's chance: $P (B ∣ A) = P (B) .$ The cyan height is the same whether we are inside the amber strip or outside it.

WHY this is the whole idea. In Step 4 the cyan height $P (B ∣ A)$ could bend with $A$ . Independence flattens it: the cut for $B$ becomes one straight horizontal line all the way across the square, at height $P (B)$ , ignoring the amber boundary entirely. That flatness is " $A$ tells you nothing about $B$ ."

PICTURE. Compare a dependent picture (cyan line jumps at the amber edge) with the independent picture (one flat cyan line across). Only the flat one lets us multiply the same numbers.

Step 6 — Read off the answer: a clean rectangle

WHAT. With the cut flat at height $P (B)$ and the strip at width $P (A)$ , the overlap is a plain rectangle. Its area: $P (A \cap B) = P (A) P (B)$

WHY it's now just $\times$ . A rectangle's area is width $\times$ height, no bending, no conditions. Because independence made width and height not talk to each other, the area is simply the product of the two side-lengths. That is the multiplication rule, and now you can see it is nothing more than "area of a rectangle."

PICTURE. Full square, amber vertical strip $P (A)$ , cyan horizontal band $P (B)$ , their crossing rectangle shaded — area labelled $P (A) P (B)$ .

Step 7 — Edge case: what if $P (A) = 0$ or $P (A) = 1$ ?

WHAT. Check the ends. If $P (A) = 0$ the strip has zero width → the crossing rectangle has zero area → $P (A \cap B) = 0$ . Correct: an impossible $A$ can never co-occur with anything. If $P (A) = 1$ the strip is the whole square, so the crossing region is just the cyan band → $P (A \cap B) = P (B)$ . Correct: a certain $A$ adds no restriction.

WHY show it. A formula you trust must survive its extremes. Both extremes fall straight out of the area picture — no special pleading.

PICTURE. Two thin panels: a collapsed (zero-width) strip, and a full-width strip.

Step 8 — Edge case: "at least one" via the empty corner

WHAT. For independent $A_{1}, \dots, A_{n}$ , "at least one happens" is easier as its opposite: "none happen." Each $A_{i}$ fails with probability $1 - P (A_{i})$ , and (independence of the failures) they all fail together with probability the product: $P (none) = \prod_{i} (1 - P (A_{i})), P (at least one) = 1 - \prod_{i} (1 - P (A_{i})) .$

WHY. "None happen" is a single rectangle (the corner where every strip is on its failure side); "at least one" is everything else. Subtracting one clean rectangle from the total $1$ beats adding up many overlapping pieces. This trick powers the Binomial Distribution.

PICTURE. The unit square with the bottom-right "all-fail" corner shaded amber; the remaining L-shape is "at least one."

The one-picture summary

Everything compresses into one diagram: a strip of width $P (A)$ , a band of height $P (B)$ , and the amber crossing rectangle whose area is the product. Independence is exactly the statement "the band's height doesn't bend when you cross the strip's edge."

Recall Feynman retelling — the whole walkthrough in plain words

Picture a floor tile exactly one metre by one metre — that whole tile stands for "something definitely happens." Paint a vertical stripe on it; the stripe's width is how likely $A$ is. Now paint a horizontal stripe; its height is how likely $B$ is. The little patch where the two stripes cross is " $A$ and $B$ both happen," and the area of a patch is just its width times its height — so its probability is $P (A)$ times $P (B)$ . The one catch: this only works if the horizontal stripe runs perfectly flat all the way across, meaning $B$ 's chance doesn't change when you step into $A$ 's stripe. That flatness is independence. If instead $B$ 's stripe jumped higher or lower inside $A$ 's region — like drawing a second red marble from a bag with one fewer red left — then you'd have to measure the height inside the stripe, which is $P (B ∣ A)$ , and you'd multiply by that instead. Multiply width by the true local height; when they don't affect each other, that true height is just $P (B)$ , and the rule becomes the simple product.

Connections

Conditional Probability — Step 3 is its definition, drawn as re-measuring inside a strip.
General Multiplication Rule — Step 4, before independence is imposed.
Mutually Exclusive Events — Step 7: why they cannot be independent (stripes must cross).
Complement Rule — Step 8's "1 minus the all-fail corner".
Binomial Distribution — repeated independent stripes multiplied.
Bayes' Theorem — flips the same conditional picture around.