Quick answer
A fixed 1% chance per independent attempt does not guarantee a reward within 100 attempts. Your chance of getting it at least once in 100 attempts is about 63.4%, leaving a 36.6% chance of getting none. Previous failures do not improve the next attempt’s odds unless the game has a documented mechanic that changes them. These are hypothetical probability calculations, not recommendations to play or spend more. (itl.nist.gov)
1. First, check what the reward percentage actually describes
Before interpreting “1%,” identify the outcome and the unit being counted.
Does the percentage apply to:
- One specific tower, character or cosmetic?
- Any reward within a rarity category?
- Each item inside a chest?
- The entire chest or pack?
- An eligible reward after another random selection has happened?
These are different questions. A percentage for “any Legendary” is not automatically the percentage for the particular Legendary you want.
For a real example, Supercell’s published Brawl Stars support information lists a 2% chance of Legendary rarity for a random Starr Drop. It separately lists possible reward types, including resources, cosmetics and Brawlers. That rarity percentage should not be presented as a 2% chance of obtaining one particular Brawler. (support.supercell.com)
Consider a separate, hypothetical two-stage system:
- A chest has a 2% chance of entering a special reward category.
- Within that category, your desired item has a 50% chance.
The chance of your item would be 2% × 50% = 1% per chest, assuming those are the applicable conditional rules. This calculation illustrates why you need both stages of the reward description, not just the headline rarity percentage. (probabilitycourse.com)
Practical check: Write down the exact outcome you want and copy the developer’s wording beside it. If the published figure describes a broader category, do not silently treat it as your item’s probability.
2. Why a reward does not become “due” after repeated failures
For the fixed-rate example in this article, assume:
- Each attempt has a 1% chance of the desired reward.
- Attempts are independent: earlier outcomes do not change later ones.
- There is no pity counter, guarantee or changing reward pool.
Under those assumptions, an attempt after 99 failures still has a 1% chance. The same is true after 199 failures. This is the memoryless property of the geometric waiting-time model: the previous wait does not change the remaining wait’s distribution. (mathworld.wolfram.com)
The important distinction is between the next attempt and a whole group of attempts.
One attempt has a 1% chance. A group offers several opportunities, so its chance of containing at least one success is higher. But once earlier attempts have failed, those opportunities are gone; their probabilities do not accumulate as credit toward the next result. (probabilitycourse.com)
Practical check: When you catch yourself thinking “I have failed so often that the next one should work,” ask whether an official rule actually changes the next attempt’s odds. If not, the feeling is not evidence of improved odds.
3. How to calculate the chance of at least one reward
For independent attempts with the same success probability, it is easiest to calculate the opposite outcome first: no successes.
With a 1% success chance, the failure chance is 99%, or 0.99. Failing every one of (n) independent attempts has probability (0.99^n). Subtract that from 1 to find the chance of at least one success. This follows from the binomial probability model. (itl.nist.gov)
Formula
[
P(\text{at least one success})=1-(1-p)^n
]
Here, (p) is the per-attempt probability written as a decimal, and (n) is the number of attempts. (probabilitycourse.com)
Using the hypothetical fixed 1% rate:
| Attempts | Chance of at least one reward | Chance of no reward |
|---|---|---|
| 10 | 9.6% | 90.4% |
| 50 | 39.5% | 60.5% |
| 100 | 63.4% | 36.6% |
| 200 | 86.6% | 13.4% |
| 300 | 95.1% | 4.9% |
Calculated from the formula above; percentages are rounded to one decimal place.
Notice that 100 attempts do not produce a 100% chance. Adding “1% + 1%” repeatedly is not the correct calculation for “at least one reward.” A group can contain multiple successes as well as none, which the full probability model accounts for. (itl.nist.gov)
Worked example: Suppose a fictional tower-game chest has a fixed, independent 1% chance of one particular cosmetic, with no guarantee. For 30 openings:
[
1-0.99^{30}\approx0.2603
]
That is about a 26.0% chance of at least one cosmetic and a 74.0% chance of none. It does not mean the cosmetic will arrive roughly one-quarter of the way through those openings.
To repeat the calculation, enter 1 - (1 - p)^n into a calculator, then multiply the result by 100. Use 0.01 for 1%, not 1.
4. What “100 attempts on average” really means
There are two useful averages, and they answer different questions.
For independent attempts at a fixed success probability (p), the expected number of attempts until the first success, including the successful attempt, is:
[
E(\text{attempts until first success})=\frac{1}{p}
]
At 1%, this equals 100 attempts. It is an average across repeated waiting periods, not a deadline for any individual player. Some waits are shorter and others are much longer. (online.stat.psu.edu)
Separately, if you complete exactly 100 independent attempts at 1%, the expected number of successes is:
[
100\times0.01=1
]
That does not mean every group of 100 contains one success. Groups with no successes and groups with multiple successes contribute to that average. (itl.nist.gov)
This explains how “one success on average” can coexist with the table’s 36.6% chance of no success in 100 attempts.
For another perspective, the chance of at least one success first exceeds 50% at 69 attempts in this hypothetical model. The average waiting time is nevertheless 100 because long waits pull the average upward. These are calculated properties of the example, not recommended targets. (online.stat.psu.edu)
Practical check: Whenever you see “average,” ask what is being averaged: attempts until the first reward, rewards within a fixed number of attempts, or results across a system that includes guarantees.
5. How documented pity systems change the calculation
A guarantee is a game rule, not a consequence of repeated bad luck.
For a documented historical example, Blizzard’s June 22, 2017 Hearthstone announcement described a Legendary guarantee within the first 10 packs of a new set, beginning with the next expansion. That illustrates a bounded guarantee rather than an unlimited fixed-rate wait. It is historical documentation, not a substitute for checking today’s pack-specific rules. (hearthstone.blizzard.com)
When reading current reward terms, distinguish three possibilities:
- Fixed chance: The probability stays the same.
- Increasing chance: The probability rises after specified failures, sometimes called soft pity.
- Guaranteed outcome by a threshold: A qualifying reward must arrive by a stated attempt, sometimes called hard pity.
If the game changes the probability according to previous outcomes, the fixed-rate formula is no longer sufficient. You need the actual changing probabilities or guarantee conditions. (online.stat.psu.edu)
Imagine a hypothetical system with a 1% chance on attempts 1–99 and a guarantee on attempt 100 if all previous attempts failed. Its chance of success by attempt 100 is 100%. Its chance by attempt 99 remains (1-0.99^{99}). The guarantee changes the endpoint; it does not make every earlier attempt certain.
Before relying on any pity system, check:
- What is guaranteed? A rarity category, featured item or exact reward?
- What counts? Openings, individual items, eligible draws or another unit?
- What resets the counter? Any qualifying reward or only the desired one?
- Where does progress apply? One pack type, event or reward pool?
- Does progress carry over? Look for explicit terms rather than assuming it does.
If the published terms do not answer these questions, the remaining attempts to a guarantee are uncertain.
6. Common mistakes that make rare rewards look more predictable
“I have already failed 70 times, so my total odds are now high.”
Your completed failures are known outcomes. Under the fixed, independent model, the chance of succeeding in the next 30 attempts is still about 26.0%. The original chance over 100 attempts is not a promise that remains available after 70 failures. (mathworld.wolfram.com)
“Ten openings at once must be luckier.”
If a batch is merely ten independent attempts with identical rules, batching does not change its probability. A batch matters only if the developer documents a separate bonus or guarantee. (probabilitycourse.com)
“A 95% chance is basically guaranteed.”
In the hypothetical 300-attempt example, about 4.9% of groups still contain no success. High probability and certainty are different.
“Someone else got it immediately, so the published rate must be wrong.”
An immediate success is possible at 1%. So is a long wait. Neither individual outcome, by itself, establishes the underlying probability. (online.stat.psu.edu)
Use calculations to describe possible outcomes, not to justify moving your stopping point.
7. A practical checklist before playing or buying
Before committing time, currency or money, make a short reward note:
- Game and platform: Identify the version and event when relevant.
- Desired outcome: Name the item, not just its rarity.
- Published probability: Record whether it is per item, opening or category.
- Changing mechanics: Record guarantees, resets and carryover terms.
- Evidence quality: Separate official documentation from player estimates.
- Stopping limit: Decide your time and spending limits before opening anything.
For parents and purchasers, the ESRB’s “In-Game Purchases (Includes Random Items)” notice identifies purchases involving randomized outcomes. Its explanation includes purchases made with virtual currency that can itself be bought with real money. The notice identifies the purchase feature; it does not provide the odds of obtaining a particular reward. See the ESRB’s explanation of randomized purchases.
Review device purchase controls and decide whether purchases should require adult approval. Keep earned currency and purchased currency clearly distinguished in your own records.
A useful final question is: “Would I still be comfortable with this decision if I received no desired reward?” If not, stop rather than treating a percentage as reassurance.
The takeaway: A fixed, independent 1% chance means a 1% chance on each attempt—not a reward owed after 100 tries. A real guarantee must come from documented game rules. Understanding either system helps you interpret rewards; it does not tell you how much to spend.