Gaming & Computers

Gacha Pull & Wish Probability Calculator

Calculate the cumulative chance of pulling a featured 5-star character using pity, soft pity and 50/50 math for Genshin, Star Rail, ZZZ and Wuthering Waves.

Banner & Pity Setup

Number Format Region

Gacha Game System

Available Pulls / Wishes

pulls

Current Pity Counter

pity

50/50 Guarantee Status

Active rules: 0.60 % base rate, soft pity from pull 74, hard pity guarantee at pull 90. The pity counter is clamped to 89 so it can never exceed the hard cap and break the cumulative math.

Probability to Get Featured 5★ Character

54.98 %

100.00 % chance of any 5★ across 90 pulls · unresolved 50/50

Pull Breakdown

Wishes Unto Hard Pity Guarantee90 pulls

Current Per-Pull 5★ Rate0.60 %

Chance for Multiple Copies (C1/E1+)18.85 %

Expected 5★ Count in Budget1.21

Average Pulls for Statistical Success76 pulls

Featured 5★ Probability54.98 %

Soft Pity Probability Curve

Per-pull rate Cumulative chance Soft pity zone (pull 74+) Your pity (0)

The shaded amber band marks the soft pity zone where the per-pull rate ramps from 0.60 % toward a guaranteed drop at pull 90. The dashed line is your current pity position.

Rates are community-datamined models of published gacha systems. Every pull is an independent random draw, so these figures describe long-run frequencies and never predict an individual wish.

What is the exact mathematical probability of winning a 50/50 gacha pull in Genshin Impact with 90 wishes accumulated?

Ninety wishes from zero pity always produce at least one 5-star, because pull 90 is the hard pity guarantee. The interesting number is not the 5-star chance but the featured chance: a single guaranteed 5-star still faces the 50/50 split against the standard character pool, so the promoted unit lands roughly 50 % of the time. The probability edges slightly above that because a lucky early drop leaves enough wishes for a second 5-star, and each additional 5-star gets its own independent 50/50 roll. Once you have already lost a 50/50, the guarantee flag turns the same 90 wishes into a 100 % featured outcome.

How does the soft pity system affect cumulative probability curves in character banners step-by-step?

The advertised 0.6 % rate is only the flat portion of the curve. From the soft pity pull onward the per-pull chance ramps steeply so that it reaches 100 % exactly at the hard cap. The cumulative curve therefore stays shallow for seventy pulls and then turns almost vertical over the following ten. That geometry explains the community wisdom that wishes 74 through 80 are the highest-yield stretch of any banner and why saving a partial pity counter between banners has real expected value.

p(n)          = base                       for n < soft pity
p(n)          = base + (1 - base) x (n - soft + 1) / (hard - soft)
p(hard)       = 1.00 (guaranteed 5-star)
P(no 5-star)  = product of (1 - p(n)) across every pull spent
P(at least 1) = 1 - P(no 5-star)
P(featured)   = sum over k of P(k five-stars) x (1 - 0.5^k)
P(featured)   = P(at least 1) when the next 5-star is guaranteed

How many wishes does each standard primogem package convert into and what success rate does it buy?

The simulation grid below assumes the standard 0.6 % base rate, soft pity at pull 74, hard pity at pull 90, a starting pity of zero and an unresolved 50/50. Notice how little a 60-pull budget buys and how sharply the numbers jump between 60 and 100 wishes as the soft pity ramp engages.

Wishes spent	At least one 5★	Two or more 5★	Featured character
20 pulls	11.34 %	0.64 %	5.83 %
40 pulls	21.39 %	2.41 %	11.32 %
60 pulls	30.31 %	5.07 %	16.49 %
80 pulls	92.53 %	10.25 %	48.99 %
100 pulls	100.00 %	26.90 %	57.19 %
160 pulls	100.00 %	97.46 %	77.42 %

Should you keep pulling once you pass the soft pity threshold or stop and save wishes for a future banner?

Stopping mid soft pity is never a loss, because pity carries forward to the next banner in every major system. The strategic mistake is the opposite one: spending high-value soft pity pulls on a banner whose featured unit you do not want, which converts a near-certain 5-star into a standard-pool character. If you are seventy pulls deep and the banner you actually want is two weeks away, holding the counter is usually the stronger play.

How do Wuthering Waves and Zenless Zone Zero pity rules compare with the Genshin Impact wish system mathematically?

Wuthering Waves is the most generous of the three: an 0.8 % base rate, soft pity around pull 65 and a hard cap at 80 push the expected cost per 5-star down near 55 pulls. Zenless Zone Zero mirrors the Genshin structure with a 0.6 % base and a 90-pull cap, shifting soft pity a single pull earlier to 73. Across a full account lifetime the difference compounds: a Wuthering Waves player reaches the same number of 5-stars for roughly ten percent fewer pulls than a Genshin player.

Frequently Asked Questions

Starting from zero pity, 90 wishes guarantee a 5-star because pull 90 is hard pity, so the chance of at least one 5-star is exactly 100%. The featured character, however, still passes through the 50/50 check, so the probability of walking away with the promoted unit is 50% from a single guaranteed 5-star, rising above that only if the 90 wishes produce a second 5-star. Accounting for the small chance of two 5-stars inside 90 pulls, the realistic featured-character probability sits near 51 to 53 percent from a fresh 50/50 state.

For the first 73 pulls the per-pull rate stays flat at the advertised 0.6%, so the cumulative curve climbs almost linearly and only reaches roughly 36% by pull 73. From pull 74 the rate ramps by about six percentage points per pull, so pull 74 is near 6.6%, pull 80 near 43%, and pull 88 above 80%. The cumulative curve therefore looks like a shallow slope followed by a near-vertical wall, which is why experienced players describe wishes 74 to 80 as the real high-yield zone of any banner.

Pity is carried between banners in every major system, so the pulls you spend early are never wasted; they simply advance the counter. Mathematically it makes no difference whether you spend 70 wishes now and 20 later or all 90 at once, because the per-pull rate depends only on the pity counter. What does matter is banner timing: spending pity on a banner you do not care about converts high-value soft-pity pulls into a standard 5-star you did not want.

Integrating the soft-pity curve gives an expected value near 62 pulls per 5-star in a 0.6% base, 74 soft, 90 hard system. Because the featured character passes a 50/50, the expected cost of a specific promoted unit is roughly 124 pulls, and the worst case of a lost 50/50 followed by a hard-pity guarantee costs 180. Wuthering Waves with an 0.8% base and an 80-pull hard cap averages closer to 55 pulls per 5-star.

In a pure 50/50 system, two consecutive losses occur 25% of the time, but Genshin Impact and Honkai Star Rail apply a hard guarantee after a single loss, so consecutive featured losses are impossible on character banners. Zenless Zone Zero uses the same guarantee model. Some titles apply a capture-radar style mechanic that lifts the effective featured rate above 50%, which slightly reduces the long-run cost of any promoted unit.

The calculation is a Markov chain over the pity counter: each state tracks the current pity and how many 5-stars have already dropped, and every pull moves probability mass either into the next pity value or back to pity zero with the counter incremented. Summing the states with two or more successes at the end of the pull budget gives the multi-copy chance. This exact enumeration is far more accurate than a naive binomial approximation, which ignores pity entirely and badly underestimates results near the hard cap.

No. Every pull is an independent draw from the rate defined by your current pity counter, so no tool can predict a specific result. What a calculator does provide is the exact distribution of outcomes over many pulls, which is the only rational basis for budgeting primogems, polychromes or astrite before a banner ends. Treat the percentages as long-run frequencies, never as a promise about any individual wish.

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