Gacha Pity Calculator

Gacha games use randomized loot systems where players spend in-game currency (often purchased with real money) for a chance to obtain rare characters or items. The pity system guarantees a high-rarity pull after a set number of attempts, preventing indefinite bad luck.

Key formulas:

Probability of NOT getting the item in n pulls: P(fail all) = (1 − base_rate)^n

Probability of getting the item at least once in n pulls: P(success) = 1 − (1 − base_rate)^n

Expected pulls to hit soft pity (varies by game): Expected Cost = (1 ÷ base_rate) pulls on average (geometric distribution)

Currency cost: Real Money Cost = Pulls Needed × Currency per Pull × Real Money per Currency Unit

What each variable means:

• Base Rate — base probability of getting the target item per pull (e.g., 0.6% in Genshin Impact for a 5-star).
• Soft Pity — a threshold where the rate increases significantly (e.g., pull 74 in Genshin).
• Hard Pity — the guaranteed pull count (e.g., 90 in Genshin, 80 in HSR for most banners).
• 50/50 — many games have a 50% chance the pity pull is the featured item; otherwise it’s a random rate-up item.

Worked example (Genshin Impact): Target: featured 5-star character. Base rate: 0.6%. Hard pity: 90. 50/50 chance.

Expected pulls for guaranteed win (worst case 50/50 loss then hard pity again): = 90 (first hard pity) + 90 (second guaranteed) = 180 pulls worst case

At $1.60 per pull average: Cost = 180 × $1.60 = $288 worst case

Expected cost (average): ~80 pulls with soft pity accounted for ≈ $128 average.