Why intuition fails
The natural instinct when you see "1% drop rate" is to multiply: one percent times a hundred packs equals one hundred percent, job done. That calculation is what statisticians call the expected value (the long-run average), and it's genuinely useful. But it is not your chance of pulling the Blook at least once. It's a different question with a different answer.
The right question is: what's the probability I get zero hits across all these packs, and then flip it? That gives you the cumulative chance of at least one hit. And because probabilities multiply when events are independent, the answer is not a straight line.
P(at least one in N) = 1 − (1 − p)NThe shape of the curve
Plot the cumulative chance against pack count and you get a curve that's steep at first, bends sharply around the 50% mark, and then asymptotes toward but never touches 100%. For a 1% drop rate, the first 20 packs take you from 0% to 18%. The next 20 take you from 18% to 33%. The next 20 from 33% to 45%. Each additional pack adds less chance than the one before.
This is called a geometric distribution, and it shows up anywhere you repeat an independent trial with a fixed success probability. Slot machines, radioactive decay, basketball free-throw streaks: same shape.
| Packs opened (p = 1%) | Cumulative chance |
|---|---|
| 10 | 9.6% |
| 50 | 39.5% |
| 69 | 50.0% |
| 100 | 63.4% |
| 200 | 86.6% |
| 230 | 90.0% |
| 459 | 99.0% |
| 1,000 | 99.996% |
Where the curve matters
Three points on the curve are worth caring about in practice. The 50% mark (coin flip on the pull), the 90% mark (insurance against an unlucky run), and the 99% mark (when you really can't afford to miss). The 50/90 guide covers how to use these as a budgeting shorthand; the threshold calculator computes them for any drop rate you plug in.
These thresholds are not spaced evenly. Going from 0 to 50% takes a certain number of packs; going from 50% to 90% takes about 3.3× that; going from 90% to 99% takes about 2× the 90% number. The cost of each additional percentage point of confidence climbs fast near the top.
Why you never hit 100%
The cumulative chance formula has an asymptote at 100%. No matter how many packs you open, the probability approaches but never quite reaches certainty, because there's always some nonzero chance the next pack (and the one after) comes back empty. This is not a bug, it's fundamental to how independent probabilities compose. The only way to get "guaranteed" would be a game mechanic that tracks misses and forces a pull after some threshold (a pity timer). Blooket does not have one.
In practice, the distinction between 99.9% and 100% doesn't matter. 99% is where most people budget because the remaining 1% of unlucky runs is small enough to absorb emotionally. Above that you're spending a lot of tokens to eliminate a tiny tail risk.
What this means for you
Three takeaways:
- Multiply-and-done is wrong. "1% × 100 packs = 100% chance" isn't how independent probabilities work. Use the exponential formula or the calculator.
- Budget to a threshold, not to the expected value. The expected value lands around 63% confidence. If you want 90%, you need more packs, about 2.3× the expected value.
- Don't chase certainty. The last 10% of confidence doubles your cost. Decide how unlucky you can afford to be, and stop there.