This calculator employs a common statistical model (Expected Value iteration) used in gacha game analysis to provide an estimated number of pulls required. The model includes the Arknights standard pity rate system.
Welcome to the **Arknights Recruit Expected Pulls Calculator**. This tool helps you estimate the average number of Headhunting pulls (rolls) needed to acquire a specific 6-star operator, taking into account the standard 2% base rate and the cumulative pity system that starts after 50 pulls.
Arknights Recruit Calculator
Estimated Expected Pulls (EP):
N/AClick ‘Calculate’ to see the detailed breakdown of the expected pulls.
Arknights Recruit Calculator Formula
The Arknights recruit probability model is complex, incorporating both a base probability and a step-up ‘Pity’ system. The calculation below uses an iterative Expected Value approach (Monte Carlo principle) to find the convergence point for the expected number of pulls required to hit the target operator.
Pbase, for N < Pity Start
Pbase + (N – Pity Start + 1) × Pity Increment, for N ≥ Pity Start
PTarget(N) = P6☆(N) × (Target Operator Chance / 100)
EP = ∑ [N × PTarget(N) × (1 – PCumulative(N-1))]
Source Links: GamePress Gacha Guide, Arknights Fandom Wiki: Headhunting
Variables
- Target 6-Star Base Rate (%): The default percentage chance of pulling any 6-star operator from the banner (usually 2%).
- Target Operator Chance (of 6-Star Pull, %): The specific chance that any pulled 6-star is the one you want (e.g., 50% for a single rate-up, 35% for a dual rate-up).
- Pity Start Pull (N): The pull number (typically 51) where the probability of receiving a 6-star operator begins to increase.
- Pity Increment (Per Pull, %): The percentage increase added to the base rate for every pull conducted after the Pity Start Pull (typically 2% per pull).
Related Calculators
What is Arknights Recruit Calculator?
The Arknights Recruit Calculator is a specialized statistical tool designed to estimate the cost, in terms of Headhunting currency (Pulls/Rolls), required to obtain a specific high-rarity (6-star) operator on a rate-up banner. Due to the nature of random probability, the calculation provides an Expected Value (EV), which represents the average number of pulls needed if the process were repeated an infinite number of times.
This calculator is essential for planning Headhunting pulls and managing resources like Originium and Orundum. By accurately estimating the expected cost, players can make informed decisions about whether to save for future operators or spend now. The primary components of the calculation are the banner’s base rate, the specific rate-up chance, and the unique pity system that progressively increases the 6-star probability until a guarantee is met.
How to Calculate Expected Pulls (Example)
- Define Parameters: Set the Base Rate to 2.0%, Target Operator Chance to 50.0%, Pity Start Pull to 51, and Pity Increment to 2.0%.
- Pre-Pity Probability: For Pulls 1 through 50, the chance of getting the Target Operator is $2.0\% \times 50.0\% = 1.0\%$, or 0.01.
- Pity Calculation: From Pull 51 onwards, the 6-star rate increases. At Pull 51, the 6-star rate is $2.0\% + 2.0\% = 4.0\%$. The Target Operator chance is $4.0\% \times 50.0\% = 2.0\%$.
- Iterative Summation: The calculator iteratively calculates the probability of *not* having the operator yet (Cumulative Failure) multiplied by the chance of getting them on the current pull. This value is then weighted by the pull number.
- Expected Pulls: Summing these weighted probabilities until the cumulative success rate approaches 100% yields the Expected Pulls (EV). For the given example, the result is approximately 82.9 pulls.
Frequently Asked Questions (FAQ)
What is the Arknights Pity System?
The pity system starts counting after 50 unsuccessful pulls. Beginning with the 51st pull, the 6-star probability increases by 2% for every subsequent pull. This continues until the rate hits 100% (at pull 99) or a 6-star operator is obtained, resetting the counter.
How accurate is the Expected Pulls calculation?
The calculation is statistically accurate as an Expected Value, representing the average outcome over many trials. In any single player’s case, the actual pulls required may be higher or lower, but the EV is the best mathematical estimate for resource planning.
Should I use the Target Operator Chance of 50% or 35%?
Use 50% if the banner features only one rate-up 6-star operator. Use 35% (or 33.33% if there are three) if the banner features two or more rate-up 6-star operators, as the chance is split among them.
Does this calculator work for guaranteed first 10-pulls?
No, this calculator assumes the standard 6-star rate applies throughout. The guaranteed 5-star or better within the first 10 pulls is usually ignored in long-term EV calculations but may slightly lower the initial cost in reality.