The parent note on TMR and voting logic leans on a pile of shorthand. Rather than list that shorthand now (it would just be noise before you have met it), this page introduces each symbol only at the moment we need it, building every one from nothing, in an order where each idea rests on the one before. A 12-year-old who has never seen any of this should be able to follow from line one.
Picture it. Three identical vending machines standing side by side. Press the same button on all three; each drops a can. Same machine, three bodies.
Why the topic needs it. TMR is defined as three modules. Without the idea "one copy among several identical copies", the word redundancy has no meaning.
Figure 1 — One input x (yellow) fans out to three identical modules M1,M2,M3 (blue boxes), each producing its own output y1,y2,y3 (green). Take away: the machine is the same; only the body and the label number differ.
Picture it. A box with a slot on the left (x goes in) and a slot on the right (y comes out). The box's inside is the rule f.
Since each module runs the same rule on the same input, we write each module's output as
yi=Mi(x)
Here yi is "the output of copy number i". Note the careful reading: Mi is the machine (the box), while yi is the number that comes out of it. If all copies are healthy, y1=y2=y3. If one is broken, its y disagrees.
Why the topic needs it. The voter compares y1,y2,y3. Everything downstream is a comparison of these three outputs, so we must first name them.
Picture it. A bar of length 1. Colour the fraction R green (works) and the remaining 1−R red (fails). The two pieces always add to the full bar.
Figure 2 — The certainty bar has total length 1. The green slice is R (module works); the red slice is 1−R (module fails). Take away: work and fail are the only two outcomes, so their probabilities must add up to exactly 1.
Why the topic needs it. The whole point of TMR is to raise reliability. To prove it helps we must measure R before and after — so R and 1−R are the currency of the entire derivation.
Picture it. Toss two coins. Chance of heads-then-heads is 21×21=41, because only one of the four equally-likely corners of a 2×2 grid is "both heads". Replace 21 by R and you get R2; for three copies, R3.
Why the topic needs it. The parent's Step 1, "P(all 3 work)=R3", is only true because failures are independent. This assumption is fragile in real life — when it breaks you get Common-Cause Failures, which the topic warns about.
Picture it. Three switches; you must turn exactly two ON. List them: (1,2), (1,3), (2,3). Three pictures, hence the number 3.
Why the topic needs it. "Exactly 2 of 3 work" can happen in 3 distinct arrangements, each with probability R2(1−R). So the total is 3R2(1−R) — the parent's Step 2. Without the counting factor you would undercount by three times.
We now have two pieces: R3 (all three work) and 3R2(1−R) (exactly two work). A TMR system is healthy whenever at least two modules work, which is "all three work" or "exactly two work". To combine an "or", we need one more rule of probability.
"All three work" and "exactly two work" are disjoint: you cannot have all three working and exactly one failing in the same trial — the count of survivors is either 3 or 2, never both. So we are allowed to add them.
Sanity check. With R=0.9: RTMR=0.81×(3−1.8)=0.81×1.2=0.972 — better than a single module's 0.9. This is the exact formula the parent note uses; you now know where every term comes from.
Picture it. A slide that starts full-height and curves gently downward, never quite touching the floor. Steeper slide = bigger λ.
Figure 3 — Reliability over time, R(t)=e−λt, for a slow-failing module (blue, small λ) and a fast-failing one (red, large λ). Both start at R=1 (yellow dot) and curve toward 0 without ever touching it. Take away: bigger λ = steeper drop = shorter useful life.
Why this tool and not a straight line? A straight line would eventually cross below zero (impossible for a probability) and would assume things fail at a constant clock time. e−λt instead says "in every equal time slice, the same fraction of survivors fails" — the natural model for random hardware death. That is exactly why the parent writes R(t)=e−λt.
Up to now Mi meant a whole machine and yi meant its output number. To talk about the voter circuit cleanly, we zoom into the simplest case where each output is a single bit — just 0 or 1. Call these three bits ==b1,b2,b3==, where bi is module i's one-bit output (so bi=yi when the output is already a single bit).
Read the voter as a sentence.V=b1b2+b2b3+b1b3
"V is 1 if bits 1-and-2 both agree on 1, OR bits 2-and-3 do, OR bits 1-and-3 do." Any pair agreeing on 1 forces V=1 — that is exactly "at least two out of three".
Figure 4 — The majority voter: three AND gates (yellow), one per pair of modules, feed a single OR gate (green) that outputs V. Take away: V=1 whenever any one pair agrees on 1, i.e. whenever at least two of the three inputs are 1.
Why the topic needs it. This is the digital voter that produces V. For continuous (analog) outputs the same "ignore the odd one out" job is done by the median — defined next.
The diagram below is a dependency graph: read it top-to-bottom, each box is an idea from this page, and each arrow means "the idea at the tail must be understood before the idea at the head". Follow any path of arrows and you are walking the exact order in which we built things; every path eventually funnels into the "TMR topic" box.
