Loot box odds: how likely is the top prize?
A 1 per cent chance per box does not mean a sure hit after 100 tries. Work out how the single chance and the overall chance fit together.
A 1 per cent chance per loot box does not mean you will get the item you want within 100 openings. If every try is independent and the chance stays the same, the probability of at least one hit after 100 openings is about 63.4 per cent. That is not a guarantee.
All figures on this page are worked examples. They do not describe a particular game or any real drop rate. For a specific game you need its published rules, especially on guaranteed rewards, changing odds and duplicates.
Start with the chance of missing
Call the chance of a hit on one try p. At 1 per cent, p = 0.01. The chance of no hit on one try is then 1 − p, which is 0.99. If the tries do not influence each other, you multiply these chances of missing together.
After n tries the probability of not a single hit is therefore (1 − p) to the power of n. What you want is the opposite event, at least one hit. The formula is:
P(at least one hit) = 1 − (1 − p)ⁿ
The formula does not distinguish between exactly one hit and several. Both count as "at least one". If you are trying to complete a whole collection, that is a different problem, where duplicates and the number of different items matter.
Example with 1 per cent per try
| Openings | At least one hit | No hit |
|---|---|---|
| 10 | 9.6% | 90.4% |
| 50 | 39.5% | 60.5% |
| 100 | 63.4% | 36.6% |
| 200 | 86.6% | 13.4% |
Even after 200 tries this model leaves a noticeable chance of never getting the item. The rising overall chance says nothing about the next single try being better: with independent openings it stays at 1 per cent.
Why "100 on average" is not a promise
With a constant 1 per cent chance, the expected number of tries until the first hit is 1 / 0.01 = 100. This expected value sums up a very large number of imagined repeats. A single run can end straight away or go on much longer.
People also confuse the expected number of hits after 100 tries with a guaranteed reward. The expected value here is 100 × 0.01 = 1. Even so, zero, one or several hits are all possible. An average does not describe what any one run will contain.
When the simple maths does not fit
- A guarantee after a fixed number of openings (often called pity) changes the distribution.
- If the chance rises after misses, p is not constant.
- If items are handed out without replacement, later chances depend on earlier results.
- If a percentage refers to a rarity tier, the chance of the exact item you want can be smaller.
So check first which event a published number actually describes. Without that, even a correctly used formula will not give a fitting answer. A new event or season can also come with different conditions from the last one.
Compare it with a simple example
In the dice calculator, "at least one six" follows the same idea of the opposite event. There the assumptions are clear: a fair six-sided die and independent rolls. The maths explains chance; it is not a recommendation to buy more paid tries.
If you would rather puzzle without any purchases, play Sudoku straight in your browser. More maths explained simply is in our odds section.