Lesson 2: BOTECs — the tool
A back-of-the-envelope calculation (BOTEC) is a rough quantitative model that makes every assumption explicit, even when the numbers are uncertain.
At the centre of every BOTEC is one bit of maths: expected value. Expected value is the probability-weighted average of all possible outcomes. If a campaign has a 30% chance of helping 34 million chickens and a 70% chance of helping zero, its expected value is:
(0.30 × 34 million) + (0.70 × 0) = 10.2 million chickens
This is the core logic of hits-based giving. A grant with a 10% chance of enormous impact can be better than a grant with a 90% chance of modest impact — if the impact when it hits is large enough.
Expected value lets you compare options that look very different:
A 100% chance of helping 1,000 chickens
A 10% chance of helping 100,000 chickens
A 1% chance of helping 10 million chickens
Their expected values are 1,000, 10,000, and 100,000 chickens — so on expected value alone, the longshot beats the sure thing by a factor of 100. That's hits-based giving in a single example: the bet you'll almost certainly lose can still be the best bet on the table.
Which is exactly where the caveat bites. A funder making this call once, who can't stomach a 99% chance of achieving nothing, might still take the safe 1,000 — and that isn't irrational. Across a hundred independent decisions, though, the longshot strategy pulls far ahead. Expected value gives you the common currency; how much weight you put on it depends on your tolerance for variance and how many bets you get to make.
Every expected value needs two kinds of numbers as inputs: values (how much is at stake) and probabilities (how likely you are to get it). The rest of the week is about how to produce numbers you'd defend for each.
That's the engine. A full BOTEC wraps a few more terms around it — how long the impact lasts, how much of the outcome you can fairly claim as yours, what it all costs — to turn this raw expected-value number into something you'd actually act on.
But right now, the thing to notice is: the BOTEC needs two kinds of numbers as inputs — values and probabilities. The rest of the week is about how to produce numbers you’d defend for each.