How does the chain rule factor P(landed, started, finished) for a three-step signup funnel?

## Where the rule comes from The only definition in play is conditional probability. For any events with `P(B) > 0`, ``` P(A|B) = P(A and B) / P(B) ``` Rearranged, that is the **multiplication rule**: `P(A and B) = P(B) * P(A|B)`. The chain rule is nothing more than applying it twice. Write the triple intersection as a pair, `(L and S)` together with `F`: ``` P(L and S and F) = P(L and S) * P(F | L and S) = P(L) * P(S|L) * P(F | L and S) ``` For `n` events the same telescoping gives ``` P(A1 and A2 and ... and An) = P(A1) * P(A2|A1) * P(A3|A1,A2) * ... * P(An | A1,...,A(n-1)) ``` Every conditioning set must have positive probability, otherwise the conditional in that factor is undefined. ## Reading it as a funnel Suppose a marketing session can *land* on the pricing page (`L`), *start* the signup form (`S`), and *finish* it (`F`). Analytics tools report exactly the chain-rule factors, one per row: - `P(L) = 0.25` — a quarter of sessions reach the page. - `P(S|L) = 0.30` — of those that reach it, 30 percent open the form. - `P(F | L and S) = 0.60` — of those that reach it *and* open the form, 60 percent complete. The end-to-end conversion is the product: `0.25 * 0.30 * 0.60 = 0.045`, or 4.5 percent of sessions. The denominators are the whole point. `P(S|L) = 0.30` is not 'thirty percent of all sessions start the form' — it is thirty percent of the *survivors* of step one. A candidate who multiplies three unconditional rates has silently assumed independence and will get a number that is wrong, usually badly. If in reality only 7.5 percent of all sessions ever start the form (`P(S) = 0.075`), then multiplying marginals `0.25 * 0.075 * ...` double-counts the landing step. ## The rule assumes nothing A common misconception is that multiplying probabilities requires independence. It does not. The chain rule is an identity, true for every joint distribution, because each factor is defined so as to make it true. Independence is the *special case* where `P(S|L) = P(S)` and `P(F|L and S) = P(F)`, and only then does the joint reduce to the product of marginals. Interviewers like this question precisely because it separates candidates who memorised 'multiply for and' from candidates who know which version of 'multiply' applies when. ## Ordering and nesting The factorisation is not unique. Any ordering of the events gives a valid chain, and all of them evaluate to the same joint probability: ``` P(L and S and F) = P(F) * P(S|F) * P(L | F and S) ``` is equally correct, just less natural to measure. Choose the ordering whose conditionals you can actually estimate — in a funnel, that is chronological order. Funnels usually have a **nesting** structure: you cannot finish the form without having started it, and you cannot start it without having landed. When `F` is a subset of `S`, which is a subset of `L`, two simplifications follow. First, the joint collapses: `P(L and S and F) = P(F)`, because finishing already implies the other two. Second, the last conditional loses a redundant term, `P(F | L and S) = P(F|S)`, since conditioning on `S` already implies `L`. That is why a funnel report can be read either as three conditional rates or as one end-to-end rate — they are the same number by the chain rule. ## Where the chain rule earns its keep The factorisation is the standard way to make a large joint distribution tractable: rather than estimate one probability over all combinations of many variables, you estimate a sequence of conditionals, each with a manageable conditioning set. It is also the backbone of sequential reasoning generally — any process described as 'this happens, then given that, this happens' is a chain-rule expression waiting to be written down. ## Common failure modes - **Multiplying marginals.** The single most frequent error; it is only valid under full independence. - **Wrong denominator.** Reporting `P(F and S and L)` as the completion rate 'among people who saw the form' when it was computed over all sessions. - **Conditioning on a zero-probability event.** `P(A|B)` is undefined when `P(B) = 0`; if a funnel step has no survivors, the next rate does not exist rather than being zero. - **Assuming the conditionals are stable.** The rule is exact for one fixed distribution; if the traffic mix shifts between the periods used to estimate different factors, the product is not a coherent joint.