MQ Walkman
Terrarian
Bethany said:Maybe eventually I'll be lucky enough to get a Lucky Coin, the Slime Staff's best friend.
Except for banners, Terraria's drops are determined by chance, like dice rolls. It is therefore impossible to work out the number of mobs you'll have to kill before being sure of getting a rare drop — it's random. But, using probability and logarithms, we can determine the number of kills at which it becomes more likely the item will have dropped.
For example, on Expert mode, killing 35 Fire Imps will mean it's more likely you'll have attained an Obsidian Rose (drop chance 2%) than you haven't. Killing 228 Imps raises the chance you've acquired an Obsidian Rose to 99%.
| Rare Item | Icon | Drop Chance (Expert Mode) | Uses | Obtainment Chance > 50% | Obtainment Chance > 90% | Obtainment Chance > 99% |
|---|---|---|---|---|---|---|
| Moon Charm |
| 4.3478% (1/23) | Late-game stat boost. | 16 kills | 52 kills | 104 kills |
| Binoculars | 
| 3.3333% (1/30) | Shifts camera’s position. | 21 kills | 68 kills | 136 kills |
| Ankh Charm Ingredients* | 
| 2% (1/50) | Mid-game immunity boost. | 35 kills | 114 kills | 228 kills |
| Money Trough | 
| 1% (1/100) | Portable, platform-less, pecuniary piggy. | 69 kills | 230 kills | 459 kills |
| Plumber’s Hat | 
| 0.4% (1/250) | Mario impersonation. | 173 kills | 575 kills | 1,149 kills |
| Wisp in a Bottle | 
| 0.3745% (1/267) | Let there be light. | 185 kills | 614 kills | 1,228 kills |
| Rod of Discord | 
| 0.2% (1/500) | Perhaps the most prized rare item in Terraria. | 347 kills | 1,151 kills | 2,301 kills |
| Discount Card, Pirate Staff | 
| 0.05% (1/2000) | 20% discounts store-wide. | 1,386 kills | 4,605 kills | 9,209 kills |
| Biome Keys | 
| 0.04% (1/2500) | Late-game gear boost. | 1,733 kills | 5,756 kills | 11,511 kills |
| Lucky Coin | 
| 0.025% (1/4000) | For use with Slime Staff. | 2,773 kills | 9,210 kills | 18,419 kills |
| Slime Staff | 
| 0.0143% (1/7000) | For use with Lucky Coin. | 4,852 kills | 16,117 kills | 32,234 kills |
| Coin Gun | 
| 0.0125% (1/8000) | Where to use those 20% discounts. | 5,545 kills | 18,420 kills | 36,840 kills |
* For the Megaphone and Fast Clock, drop chance depends on monster killed (Pixies or Dark Mummies – 1%).
These values are best derived using drop chance complements. Instead of considering the increasingly complex tree of successful drops, we consider only unsuccessful drops. Taking the Obsidian Rose example:
P(Drop) = 2%
Therefore,P(noDrop) = P'(Drop) = 100% – 2% = 98%
Repeatedly not getting an Obsidian Rose multiplies this value by itself: 98% × 98% × 98% × … × 98%. There is a 94.11% chance, for example, of not getting an Obsidian Rose after killing 3 Fire Imps. Let n be the number of times required for the event, P(noDrop), to take place before becoming as likely as the complement of the same number of times; thus:[P'(Drop)]ⁿ = 1 – [P(noDrop)]ⁿ
2[P'(Drop)]ⁿ = 1
[P'(Drop)]ⁿ = 0.5
Substituting values (the cool bit):0.98ⁿ = 0.5
n = log_0.98 (0.5)
n = ln(0.5) / ln(0.98)
n = 34.3096184915
0.3096 of an Imp is meaningless, so we round up to 35 Imps. Sure enough:1 – (0.98)^34 = 49.6%
1 – (0.98)^35 = 50.7%
And it rolls over to be less likely not to have the Obsidian Rose, as expected.
Aside from finding it interesting, I think this data is useful in terms of tempering your expectations. If you want to start a Summoner playthrough and you need the Slime Staff; statistically, you can expect a long wait. But remember these two things:
- “Coins don't have memory.” Do not sue me if, upon killing your 32,234th slime, the Staff is nowhere to be seen. It's possible. It's possible to kill 100,000 slimes and suffer the same fate (0.0000624% chance). Terraria isn't mathematically obliged to give you anything.
- Maths aside, it's still gambling. Sometimes, you'll just get lucky!
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– so I've manually redone the Expert one.
And, in the meantime, may the 
