P(at least one in N) = 1 − (1 − p)ⁿDefinition
Cumulative probability is the chance of an event happening at least once across N independent trials. In Blooket: the chance of pulling your target Blook in any of N packs you open.
It is not the same as multiplying the drop rate by the number of packs. A 1% drop rate × 100 packs is not a 100% chance, it's about 63.4%. Multiplication gives the expected count, not the probability of at least one hit.
The formula
Where p is the per-pack drop rate (as a decimal, 1% = 0.01) and N is the number of packs:
Compute the chance of missing every pack, that's (1 − p)^N. Then subtract from 1 to get the chance of at least one hit.
Worked examples (1% Legendary)
Plug p = 0.01 and a few values of N into the formula:
- 10 packs → 1 − 0.99¹⁰ = 9.6%.
- 50 packs → 39.5%.
- 69 packs → 50.0% (the 50% threshold for a 1% drop).
- 100 packs → 63.4% (the famous "1 − 1/e" constant).
- 230 packs → 90.0% (the 90% threshold).
- 459 packs → 99.0% (the 99% threshold).
Why you never hit 100%
The curve 1 − (1 − p)^N asymptotes toward 100% but never reaches it. Each additional pack subtracts a smaller and smaller slice of the remaining probability mass. There is no pity timer in Blooket, every pack opening is genuinely independent, so you can't guarantee a pull no matter how many packs you open.
In practice, the 99% threshold is where most token-budget calculations stop. The remaining 1% of unlucky runs is small enough to absorb, and chasing 99.9% costs roughly double the tokens for negligible additional confidence.
Why 50, 90, and 99 are the standard thresholds
Cumulative probability is a continuous curve, but in practice three points on it carry almost all the planning weight: 50% (coin-flip confidence), 90% (strong insurance), and 99% (near-guaranteed).
These thresholds correspond to the multipliers 0.7 / p, 2.3 / p, and 4.6 / p where p is the per-pack drop rate. For a 0.5% Legendary, that's 139 / 460 / 920 packs. For a 0.02% Chroma, it's 3,500 / 11,500 / 23,000 packs. The 99% threshold is roughly 6.6× longer than the 50% threshold for any drop rate, the cost of going from coin-flip to near-certainty is steep no matter how rare the target.
Calculators on this site default to showing all three. If you only have a budget for the 50% threshold, you have a 1-in-2 shot of pulling the target. Most token-budget planning lands at 90% as the practical sweet spot, the 99% threshold costs ~2× more tokens than the 90% threshold for one-tenth the marginal confidence gain.
Approximations for small drop rates
When p is small (typical for Blooket pack drops), 1 − (1 − p)^N ≈ 1 − e^(-pN). This is the Poisson approximation, and it gets within 1% accuracy for p ≤ 0.05 and any N. The exponential form is easier to invert: solving 1 − e^(-pN) = c gives N = -ln(1 − c) / p, which is where the 0.7 / p, 2.3 / p, 4.6 / p multipliers come from.
For very large N (Chroma chases at 0.02%), the cumulative probability plateaus near 100% slowly. Each additional pack adds less and less, this is why doubling your token budget at 90% confidence does not double your effective progress, it eats into a smaller and smaller slice of remaining probability.
The expected value 1/p is sometimes mistaken for the 50% threshold. They're different. 1/p is the mean number of packs to one hit (also the inverse of the rate). The 50% threshold is the median, the pack count at which half of all chase runs would have already succeeded. For exponentially-distributed waiting times, median = ln(2)/p ≈ 0.693/p, while mean = 1/p. The median is always smaller because the long tail of unlucky runs pulls the mean up.
Common misconceptions
"100 packs at 1% guarantees a pull." No. The cumulative chance is 1 − 0.99^100 = 63.4%. Multiplication (drop rate × pack count) gives the expected count of pulls, not the probability of at-least-one.
"My next pack is more likely because I've missed so many." No. Each pack is independent. There is no pity timer, no streak adjustment, no compounding mechanic. The drop rate on pack 1,000 is identical to the drop rate on pack 1.
"Cumulative probability eventually hits 100%." No. The curve asymptotes toward 100% but never reaches it. There is always a non-zero chance of missing the target, no matter how many packs you open. This is mathematically baked into the formula and not a quirk of Blooket.
Frequently asked questions
What is cumulative probability in Blooket?
The chance of pulling a target Blook at least once over N pack openings. Computed via 1 − (1 − p)^N where p is per-pack drop rate.
Is there a pity timer in Blooket?
No. Each pack opening is statistically independent. Only seasonal 100%-Chroma packs (Lovely / Lucky / Spring) have a guaranteed-drop mechanic, and they're not Market packs.
What does the 50% threshold mean?
The pack count at which your cumulative probability of pulling the target hits 50%, the coin-flip point. For drop rate p, the 50% threshold is roughly 0.7 / p packs.
Why isn't 100 packs at 1% a guaranteed Legendary?
Multiplication (0.01 × 100) gives the expected count of pulls, not the probability of at least one. The actual at-least-one chance is 1 − 0.99^100 = 63.4%. Cumulative probability never reaches 100%.
How many packs do I need for 99% confidence?
About 4.6 / p. For a 0.5% Legendary, ~920 packs. For a 0.02% Chroma, ~23,000 packs. The 99% threshold is roughly 6.6× longer than the 50% threshold for any drop rate.
Can I just buy more packs to make a rare drop more likely?
Yes, cumulative probability grows monotonically with N. But it grows with diminishing returns, each additional pack adds less to your total chance than the one before. Eventually, more packs is paying full price for almost no gain.
See also
A Legendary is the second-rarest Blook tier in Blooket, rarer than Epic, more common than Chroma, with drop rates between 0.2% and 1% per pack.
A Chroma is the rarest obtainable Blook tier in Blooket. Per-pack drop rates sit between 0.01% and 0.08%, and only specific market packs contain one.
A Pack is a token-priced loot box in the Blooket Market. Open one to receive exactly one random Blook, with drop rates determined by the pack's rarity table.