Variance, Dry Streaks, and Confidence: When RNG Is Actually Bad Luck
Why do two players with the same drop rate have opposite experiences—one hits early, the other farms 200 runs with nothing? The answer is variance. This guide shows how to quantify dry streaks, calculate the probability of “no drops” after N tries, and decide when your drought is still normal RNG versus statistically suspicious. We include formulas, worked examples, and links to calculators so you can validate any pity system, gacha banner, raid drop, or relic farm.
Variance is the mathematical reason dry streaks happen even when drop rates are fair. If your legendary chance per run is 5%, the probability of getting nothing after 60 runs is still (1 - 0.05)60 ≈ 4.6%. That feels awful but is statistically normal. The goal of this article is to give you a playbook for answering two questions: “Is my drought still within expectations?” and “How many more attempts until I reach 50%, 75%, or 95% confidence?”
Key concepts you need
- Per-try probability (p): The chance a single attempt succeeds (e.g., 5% drop, 0.5% pity base).
- Complement rule: P(none) = (1 - p)N. P(at least one) = 1 - (1 - p)N.
- Variance: Why outcomes spread out; EV alone cannot describe streakiness.
- Confidence checkpoints: N needed for 50%/75%/90%/95% success probability.
- Pity / bad-luck protection: Rising p after misses lowers variance and shortens dry streaks.
Formulas and quick calculator links
Use the complement rule to calculate the odds of at least one success. For a fixed drop rate p:
P(at least one) = 1 - (1 - p)^N
Solve for N: N = ln(1 - target) / ln(1 - p)With pity (e.g., +5% per miss after 200 pulls), per-try probability changes each attempt. Use our calculators to iterate each pull: RSL Shard & Mercy Calculator, Apex Pack Probability, Warframe Relic Drop.
Worked examples
- No pity, 5% drop, N = 60 runs. P(none) = (0.95)60 ≈ 4.6%, so P(at least one) ≈ 95.4%. A 0/60 drought is rare but plausible.
- Pity case: base 0.5%, +5% after 200 pulls, hard pity 220. Use the shard mercy calculator to simulate. You’ll cross 95% long before 220 because pity ramps probability sharply.
- Gacha banner, 0.6% 5-star, 90 pity. At 80 pulls, P(at least one) ≈ 1 - (0.994)80 ≈ 40%. Dry streaks near soft pity are common.
When is a drought “abnormal”?
There is no single cutoff, but community standards use 5% or 1% tails. If P(none after N) < 1%, the drought is extreme. At that point, double-check that your assumed drop rate p is correct, you tracked attempts accurately, and the event rules match your model.
- Check your assumptions: Was there a 2x event? Did you miscount attempts? Is the drop rate per party or per player?
- Run a confidence interval: If you observed k successes in N trials, the binomial CI for p may reveal your assumed rate is off.
- Use a tracker: Log attempts so you can defend or adjust your math.
Reducing variance in practice
- Raise attempts per hour: Variance shrinks when you get more trials quickly (faster runs, better routing).
- Prefer pity systems: Pity flattens tails; toggle pity calculators to see how fast risk collapses.
- Pool with teammates: In best-of-squad systems (Warframe relics), independent rolls combine to reduce drought risk.
- Stop-loss thresholds: Decide a max number of attempts (e.g., 95% checkpoint) before pausing or changing strategy.
Common mistakes
- Using EV alone—EV ignores streakiness; variance and P(any) matter for planning.
- Assuming 2x events change pity steps—most games double base rate only.
- Mixing per-run and per-item rates—raid tables often split personal vs shared loot.
- Forgetting independence—squad selection is “best-of N independent rolls,” not additive probabilities.