How Gacha Probability Works
Getting a desired item from a gacha or loot box system is not as simple as adding up percentages. Each pull is an independent event, so the probability of getting at least one success in n attempts is P = 1 - (1-p)^n, where p is the single-pull rate. For example, a 1% rate over 100 pulls gives only a 63.4% chance — not 100%. This is because each pull is independent and the probabilities compound geometrically, not linearly.
The Gambler's Fallacy
Many players believe that after a string of failures, the next pull must be "due" for success. This is the classic Gambler's Fallacy. In a fair random system, each pull is completely independent of the previous ones — 100 consecutive failures do not change the probability of the 101st pull at all. It remains exactly p. Understanding this helps set realistic expectations about how many pulls are actually needed.
Pity Systems and Expected Value
Many modern gacha games include a pity system that guarantees a rare item after a certain number of failed pulls (e.g., 90 pulls). This calculator computes pure independent probability without any pity mechanism. If your game has pity, your actual expected cost will be lower than the raw numbers suggest. The average number of pulls needed without pity is 1/p — for a 1% rate, that's an average of 100 pulls per item.
Frequently Asked Questions (FAQ)
Q. Doesn't 1% × 100 pulls = 100% success?
A. No — probabilities multiply, not add. The probability of failing all 100 pulls is (0.99)^100 ≈ 36.6%, which means the success probability is only about 63.4%. This is a common misconception.
Q. How many pulls do I need for a 90% success chance?
A. Use the formula n = ln(0.1)/ln(1-p). For a 1% rate, that's approximately 230 pulls. For 99% certainty, you'd need around 459 pulls.
Q. Can gacha rates be different from what's advertised?
A. In principle, the rates should match what's disclosed. Many regions now require game publishers to disclose gacha rates by law. If a game's rates seem off, checking community statistical analyses can provide additional insight.