🎰Gacha Probability Calculator

Why doesn't a 1% probability guarantee a hit in 100 pulls?

When pulling a 1% gacha 100 times, the probability of getting at least one hit is approximately 63.4%, not 100%. Each draw is independent, and there's a remaining probability of 100 misses in a row (0.99 to the 100th power, approximately 36.6%).

• Formula for probability of at least one hit：The probability of at least one hit is calculated as "1 − (1 − drop rate) to the power of number of pulls". At a 1% drop rate, 230 pulls reaches approximately 90%, and 459 pulls reaches approximately 99%.

What is expected value (expected pulls)?

Expected pulls is the average number of pulls needed to get your target item when repeating the same gacha many times. Expected pulls is calculated as "100 ÷ drop rate (%)"; at a 1% drop rate it's 100 pulls, at 3% it's approximately 33.3 pulls.

• Expected pulls doesn't guarantee a hit：Even if you pull a 10% gacha 10 times (the expected pulls), the probability of getting at least one hit is only approximately 65.1%.

Why does completing all items require so many more pulls?

The number of pulls to collect all items is often many times the number of item types. As you collect more items, new items become rarer, and the last item type takes the longest. This problem is known as the Coupon Collector Problem.

• Example: 10 item types with equal probability：For a gacha where 10 items each have a 10% drop rate, the expected pulls to complete all items is approximately 29.3 pulls, about 2.9 times the number of item types.
• When drop rates differ：When drop rates differ, the rarest item increases the total pulls. For 3 items with 1%, 2%, and 3% drop rates, the expected pulls is approximately 121.7 pulls.