From a list of chances to one number
A random reward, such as a chest, a pack or a crafting roll, usually pays out one of several items with different chances. To compare it with a fixed-price item, you need one number. The weighted average gives it: multiply each item’s value by its probability and add the results.
An ordinary average treats every item equally. A weighted average lets common items count more than rare ones, in proportion to how often they appear. It is the same calculation as expected value, applied to the question of what a random reward is worth.
A worked example
A chest drops one item with these chances and values (in gold):
| Item | Chance | Value | Chance × value |
|---|---|---|---|
| Common | 70% | 10 | 7 |
| Rare | 25% | 40 | 10 |
| Epic | 5% | 200 | 10 |
The weights add to 100%, which is the first thing to check. The weighted average value is 7 + 10 + 10 = 27 gold.
Notice what the table reveals. The epic item appears in only one chest in twenty, yet it contributes as much to the average as all the rare drops combined. A plain average of the three item values, (10 + 40 + 200) / 3 = about 83.3, would overstate the chest badly because it ignores how seldom the epic appears.
What a fair price means
If the chest costs 27 gold, then on average you break even. A “fair price” in this sense is simply the weighted average value. At 30 gold, the chest costs about 3 gold more than it returns, on average. At 20 gold, it returns about 7 more than it costs.
Remember that average does not mean typical. With these numbers, 70% of the time you receive something worth 10, and only 1 chest in 20 returns more than the price. A chest that is fair on average can still disappoint on most openings. This is the same lopsided pattern that creates the average outcome trap.
Where do the values come from
The weighted average is only as good as the values going into it. There are a few ways to set them.
- Market price. If items trade, use what they sell for after fees and taxes, not the listed price.
- Vendor or conversion value. If items can only be broken down or sold to a shop, that is a floor, and it can be far lower than a trading price.
- Personal value. A piece of gear you will actually use may be worth much more to you than to the market, and a duplicate of something you already own is often worth much less. Weighted averages should reflect your situation.
- Comparable fixed-price items. If the same item can be bought directly, that price caps what the random version is worth, whatever the odds.
If values are guesses, treat the result as an estimate, and test how much it changes when a value is off by 20 or 30%. The rare tiers usually matter most, so the uncertainty tends to sit there.
Extending the method
The same approach covers several related questions.
- Cost per item you want. If you only want the epic and each chest costs 27, with a 5% chance per chest the average number of chests is 1 / 0.05 = 20, so the average cost is about 20 × 27 = 540 gold. That is an average over many attempts; individual players may need far fewer or far more, as the waiting time discussion shows.
- Value after duplicates. If a duplicate is worth less than a first copy, lower its value in the table, which lowers the weighted average.
- Multiple rolls per purchase. Weighted averages add, so a pack with five independent draws is worth five times the single-draw value.
- Bonus guarantees. A guaranteed rare or better in each pack changes the probabilities, so recompute the table for the guaranteed slot separately.
Pack odds and rarity tiers, and the ways drop rates are published, vary a lot between games. Some games show clear rates and some do not, so check the source before trusting the weights. Paid random rewards in particular carry spending risk, and the average value of a paid chest is often below its price, so decide on a limit before you start and stick to it.
Common mistakes
- Using the wrong weights. Make sure probabilities add to 100% and that they describe the same draw you are valuing.
- Valuing the top prize at its sticker price. If you cannot realistically sell it, or only one buyer wants it, the value is lower than the number on the screen.
- Anchoring on the biggest item. A bright, valuable prize draws attention and can make the whole bundle feel worth more than its weighted average, a pattern linked to the anchoring effect.
- Forgetting that the average is long-run. A weighted average tells you about many openings together, and about a handful of them it says very little.
Computing a weighted average takes a few lines of arithmetic, and it is one of the clearest ways to see what a random reward really offers, whether or not you decide it is worth buying.