Gacha Probability Calculator
Free Gacha probability tool for odds & chance. Enter your details to get instant, tailored results and guidance. Includes formulas and worked examples.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
Gacha Probability Calculator
Calculator
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Formula: P(at least 1) = 1 - (1 - r)^n
Worked example — 38.2% chance in 80 pulls | 100% with pity at 90 | Expected: 167 pulls
Formula
P(at least 1) = 1 - (1 - r)^n
Where r is the drop rate (as a decimal) and n is the number of pulls. For multiple copies, the binomial distribution is used: P(exactly k) = C(n,k) x r^k x (1-r)^(n-k). Pity systems guarantee a drop within a fixed number of pulls regardless of probability.
Worked Examples
Example 1: Featured 5-Star Character (0.6% rate)
Problem:Calculate probability of getting a featured 5-star character with 0.6% rate in 80 pulls, 90-pull pity.
Solution:Drop rate: 0.6% = 0.006 Probability per pull of NOT getting: 0.994 P(at least 1 in 80) = 1 - 0.994^80 = 1 - 0.618 = 38.2% Expected pulls: 1/0.006 = 167 50% chance at: 115 pulls 90% chance at: 384 pulls With 90-pull pity: guaranteed by pull 90
Result:38.2% chance in 80 pulls | 100% with pity at 90 | Expected: 167 pulls
Example 2: SSR Card in Mobile Game (3% rate)
Problem:What are the odds of pulling at least 2 SSR cards with 3% rate in 50 pulls? Cost is $3/pull.
Solution:Drop rate: 3% = 0.03 P(exactly 0) = 0.97^50 = 21.8% P(exactly 1) = C(50,1) x 0.03^1 x 0.97^49 = 33.7% P(at least 2) = 1 - 21.8% - 33.7% = 44.5% Expected SSR cards: 50 x 0.03 = 1.5 Total cost: 50 x $3 = $150
Result:44.5% chance of 2+ SSR | Expected: 1.5 SSR | Cost: $150
Frequently Asked Questions
How does gacha probability work?
Gacha probability follows the principles of independent random events. Each pull has a fixed probability (drop rate) of yielding the desired item, and each pull is independent of previous ones. The probability of NOT getting the item in a single pull is (1 - drop rate). For multiple pulls, the probability of getting at least one copy is calculated as 1 minus the probability of failing all pulls: P = 1 - (1 - r)^n, where r is the drop rate and n is the number of pulls. This means probability increases with more pulls but never reaches 100% without a pity system. A common misconception is that probabilities add up linearly, but they actually compound multiplicatively.
What is a pity system in gacha games?
A pity system is a mechanic that guarantees a rare item after a certain number of unsuccessful pulls. For example, if a game has a pity at 90 pulls, you are guaranteed the featured item by your 90th pull if you have not obtained it naturally. Pity systems come in several forms: hard pity provides a guaranteed drop at a fixed count, soft pity gradually increases the drop rate as you approach the pity threshold, and the 50/50 system gives a chance at the featured item versus any random item of the same rarity. Some games carry over pity count between banners while others reset it. Pity systems significantly affect the expected cost and make gacha outcomes more predictable for planning purposes.
How much should I budget for a specific gacha character?
Budgeting for gacha depends on your risk tolerance. For a typical 0.6% drop rate with 90-pull pity: the median (50% chance) requires about 115 pulls. For 90% confidence, you need approximately 384 pulls. With pity, the absolute maximum is 180 pulls for a guaranteed featured character on a 50/50 system. To convert pulls to cost, multiply by the cost per pull (typically $2-3 in premium currency). So for a featured 5-star character at 50% chance: roughly $230-345, and for near-certainty: $360-540. Always set a firm budget before pulling and never chase losses. Consider saving free currency over multiple patches to reduce real-money spending.
What is the difference between rate-up and base rate?
The base rate is the overall probability of getting any item of a particular rarity tier in a gacha pull. The rate-up (or featured rate) is the probability specifically for the promoted or featured item. For example, a game might have a 1.6% base rate for any 5-star character, but the rate-up character might have a 0.5% specific drop rate within that 1.6%. When you do pull a 5-star, there is often a 50/50 chance it is the featured character versus a standard one. This distinction is crucial for calculating the true probability of getting a specific character. Some games implement an escalating guarantee where losing the 50/50 ensures the next 5-star is guaranteed to be the featured one.
How do I calculate the probability of getting multiple copies?
Getting multiple copies (for constellations, dupes, or limit breaks) requires binomial probability calculations. The probability of getting exactly k copies in n pulls with probability p per pull is: C(n,k) x p^k x (1-p)^(n-k), where C(n,k) is the binomial coefficient. For the probability of getting at least k copies, sum the probabilities from k to n. Multiple copies become exponentially more expensive. For example, with a 1% rate, getting at least one copy in 100 pulls has a 63% chance, but getting at least two copies in the same 100 pulls drops to only about 26%. This is why max-constellation or max-refinement targets can cost thousands of dollars in premium currency.
What is the difference between odds and probability?
Probability is expressed as a number between 0 and 1 (or a percentage), representing the likelihood of an event. Odds compare favorable outcomes to unfavorable ones — odds of 3:1 means 3 wins for every 1 loss, which is a probability of 3/(3+1) = 75%. Casinos often express odds differently from true probability to build in their house edge.
What is the probability of rolling a specific number on a standard die?
A fair six-sided die has 1/6 ≈ 16.67% probability for each face. Rolling at least one specific number in two rolls = 1 − (5/6)² ≈ 30.6%. Rolling two specific numbers on two dice = 1/36 ≈ 2.78%. These calculations multiply individual probabilities for independent events.
What is a fair game in probability theory?
A fair game is one where the expected value for all players is zero — no participant has a mathematical advantage. In practice, most casino games are unfair (negative EV for players) due to the house edge. Flipping a coin for even money is a fair game; flipping for $0.90 per win and $1 per loss is unfair.
What is the birthday problem in probability?
The birthday problem asks: how many people are needed for a 50% chance two share a birthday? The answer is just 23 people — surprising because there are 365 days. The probability no two people share a birthday with n people = (365/365)(364/365)(363/365)...(365−n+1)/365. With 23 people this equals ≈50.7%, meaning a shared birthday is more likely than not.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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