Expected Value, Explained With Loot Drops
If you only ever learn one idea from probability, make it this one. Expected value takes a whole reward table — every possible drop and how likely it is — and squeezes it into a single honest number: the average reward a player walks away with per pull. Once you can read it, you can compare two systems before a single player touches either.
The one-line definition
Expected value is what you'd get per pull if you could run the system forever and take the average. You compute it by walking through every outcome, multiplying what it's worth by how likely it is, and adding those pieces together.
That's the whole formula: multiply, then sum. It looks fancier written in symbols, but it's just a weighted average where the weights are probabilities.
Walking a real table
Say a chest pays out one of three things. Five percent of the time you strike it big at 100 gold, a quarter of the time you get a tidy 20 gold, and the other seventy percent you pocket a modest 5 gold. What's the chest actually worth on average?
Multiply each reward by its chance and add: 0.05·100 + 0.25·20 + 0.70·5 = 13.5 On average, every chest hands the player 13.5 gold. Not the 100 that hooks the eye, not the 5 that disappoints — thirteen and a half. That number is the truth of the table.
Why designers love it
Expected value turns arguments into arithmetic. Suppose your economy lead wants the chest to average 15 gold to match the effort it takes to open one. You're at 13.5. Now the conversation is concrete: nudge the rare tier up, or shift a little weight from the 5-gold outcome to the 20-gold one, and re-check. Every tweak has a visible effect on one number you both trust.
It also lets you compare unlike things. A daily reward, a boss drop and a crafting refund can all be reduced to their expected value in the same currency, so you can see at a glance which faucet is quietly flooding your economy.
What expected value hides
Here's the catch, and it's a big one: two tables can share an expected value and feel completely different. A chest that always pays exactly 13.5 gold and a chest that usually pays 5 but occasionally erupts into 100 have the same average — yet one is dull and the other is thrilling (or infuriating). The average tells you the centre of the story, not its shape.
That shape is called variance, and it's where the feel of a run lives. Expected value is your first tool, not your only one. Learn it, lean on it, and then reach for its partner.
A quick habit to build
Whenever you look at a reward table, compute its expected value before you judge it. It takes thirty seconds and it stops you from being fooled by a shiny top prize with a tiny chance attached. The number won't tell you everything, but it will keep you honest.
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