Foundations — Fault tree analysis (FTA) — top-down, AND - OR gates

Before you can build or read a fault tree (see the parent topic), you need a toolbox. The parent note freely uses words like probability, product, independent events, and symbols like $P$, $\prod$, $\times$. Here we earn each one from absolute zero, in an order where every item leans only on the ones before it.

1. What is an "event"?

The picture: think of a light switch. It is either ON (the event happened) or OFF (it did not). There is no "half-on".

Why the topic needs it: a fault tree is built entirely out of these on/off boxes. Every "Solar Fail", "Cell1", "HGA Fail" in the parent note is one of these switches. Before we can combine failures, we must agree that each failure is a clean yes/no thing.

2. Probability — a number between 0 and 1

The symbol $P$ is just shorthand for the words "the probability of". So $P(\text{Battery Fail})=0.02$ reads "the probability that the battery fails is 0.02".

Why the topic needs it: every basic event in the parent note ($P=0.001$, $P=0.05$, …) is a point on this ruler. The entire "quantitative analysis" is arithmetic on these numbers.

3. The complement — "the chance it does NOT happen"

The picture: the whole ruler has length $1$. If the "happens" part takes up a slice of length $P$, the leftover slice — "does not happen" — must be the rest, $1-P$.

Why the topic needs it: the OR-gate formula in the parent note is built entirely from complements — "1 minus the chance nothing fails". You cannot read that formula until $1-P$ feels obvious.

4. Independent events — when one has no effect on the other

The picture: two separate dice. Rolling a 6 on the first die does not change the odds of the second. They live in separate worlds.

Why the topic needs it: both the AND and OR probability formulas in the parent note carry a hidden footnote — "for independent events". If events are not independent, those formulas are wrong. So this concept guards every calculation.

5. Multiplication of probabilities — the "AND" of independent events

This is the single most important idea for AND gates.

The symbol $\times$ means ordinary multiplication. The picture: a rectangle. If the width is $P(A)$ and the height is $P(B)$, the area is the chance both happen.

Why the topic needs it: this is the AND-gate formula. Every AND gate in the parent note multiplies its inputs.

6. The product symbol $\prod$ — "multiply a whole list"

Reading it piece by piece: ${\prod }_{i=1}^n{P}_i=P_1\times P_2\times \cdots \times P_n$

• $i$ is a counter (like a page number). It starts at the bottom value and steps up by 1.
• $i=1$ under the symbol: start counting at 1.
• $n$ on top: stop when the counter reaches $n$ (the total number of items).
• $P_i$: "the $i$-th probability" — plug in $i=1,2,3,\dots$ in turn.

Why the topic needs it: both the AND-gate formula ($\prod P_i$) and the OR-gate formula ($\prod (1-P_i)$) in the parent note use this symbol. It is just "multiply the list" — nothing more.

7. The OR gate formula — putting it all together

Now every piece from sections 3, 4, 5, 6 combines into the parent note's scariest-looking formula.

Read it inside-out, using tools you now own:

1. $1-P_i$ — the chance input $i$ survives (complement, §3).
2. $\prod (1-P_i)$ — the chance all inputs survive at once (multiply independents, §5–6).
3. $1-(\text{that})$ — the chance they do not all survive, i.e. at least one fails (complement again, §3).

Why the topic needs it: this is how the parent computes $P_{\text{Solar}}=0.016$. You now understand every symbol in it.

8. The tree picture itself — gates, branches, top and bottom

The picture: a family tree drawn upside-down. The disaster sits on top; its causes branch downward; the smallest causes are the leaves at the bottom. Reading down = "what could cause this?". Computing up = "so how likely is the disaster?".

Why the topic needs it: this is the object the whole method produces. See Boolean logic for the algebra behind AND/OR, and Reliability Block Diagrams for the same logic drawn a different way.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.