Coupon Collector Expected Trials Calculator

What Is the Coupon Collector Problem?

The coupon collector problem is a classic question in probability theory. Imagine there are n distinct coupon types, and each time you draw a coupon, it is chosen uniformly at random from those n types. You keep drawing with replacement (putting coupons back) until you have at least one of every type. The central question is:

On average, how many draws will it take to collect all n coupon types?

Typical real-world analogues include:

• Collectible cards, stickers, or toy figurines in blind packs.
• Gacha or loot-box systems in games, where you want all characters or items.
• Randomized software tests where each distinct bug type is a “coupon.”
• Sampling from a population until every species or category is observed at least once.

This calculator takes the number of coupon types and optionally a time per draw, then returns the expected number of draws (and time) required to complete the collection under the standard coupon collector assumptions.

How the Expected Number of Draws Is Calculated

The key idea is to think of collecting coupons as happening in stages. At each stage you are waiting to see a new type that you have not collected before.

1. Stage 1: You start with no coupons. The very first draw is always a new type, so the expected number of draws for the first new coupon is 1.
2. Stage 2: Now you have 1 type in your collection and n - 1 types are still missing. The probability that any single draw gives you a new type is (n - 1) / n. The expected number of draws to get that next new coupon is 1 / ((n - 1) / n) = n / (n - 1).
3. Stage k: Once you have collected k - 1 distinct types, there are n - (k - 1) types still missing. The probability that the next draw is new is (n - k + 1) / n, so the expected number of draws to get the k-th new type is n / (n - k + 1).

To finish the collection you need to pass through all stages from k = 1 up to k = n. The expected total number of draws is therefore the sum of the expected draws for each stage:

If you re-index this sum, you obtain the famous harmonic number:

Here H_n is the n-th harmonic number. For large n, this grows approximately like ln(n) + γ, where γ ≈ 0.57721 is the Euler–Mascheroni constant. That means the expected number of draws grows a bit faster than n log n, which can be surprisingly large even for moderate n.

Worked Example: Small Collection

Suppose there are 5 coupon types in a cereal-box promotion, and you want to know how many boxes you should expect to buy to get them all at least once.

Set Number of Coupon Types = 5 in the calculator and leave the time field blank or at zero if you only care about draws.

The expected number of draws is:

E = 5 × (1 + 1/2 + 1/3 + 1/4 + 1/5) ≈ 5 × 2.2833 ≈ 11.42

Interpretation:

• On average, you need about 11 or 12 boxes to get all 5 toys.
• This is more than double the number of types because later toys are harder to find; you often get duplicates of ones you already have.

If each box takes about 30 seconds to buy and open, and you enter 30 into the Seconds Per Draw field, the expected time becomes:

Expected time ≈ 11.42 × 30 ≈ 342.6 seconds ≈ 5.7 minutes.

Worked Example: Larger Collection

Now consider a game with 100 distinct items you want to collect. Set Number of Coupon Types = 100. Suppose each draw (a pull, spin, or pack opening) takes about 5 seconds, including animations and menu navigation.

The exact expectation uses the harmonic number H_100, but it is often approximated as ln(100) + γ:

H_100 ≈ ln(100) + 0.57721 ≈ 4.6052 + 0.5772 ≈ 5.1824 E ≈ 100 × 5.1824 ≈ 518.24 draws

With 5 seconds per draw:

Expected time ≈ 518.24 × 5 ≈ 2,591 seconds ≈ 43.2 minutes.

This example illustrates how expectations can quickly become large. Even though there are only 100 items, you might need over 500 pulls on average to see every one at least once.

How to Use the Calculator

1. Number of Coupon Types
   Enter a positive integer representing how many distinct outcomes exist. Examples: the number of different sticker designs, unique monsters, or independent categories in your experiment.
2. Seconds Per Draw (optional)
   Enter the average time per trial. This could be seconds per pack opening, seconds per automated test run, or seconds per sample taken. If you leave this field empty or set it to zero, the calculator will only report the expected number of draws, not the time.
3. Run the calculation
   Click the button to compute the expected draws. If you provided a positive time per draw, the calculator will also show the expected total time by multiplying draws by seconds per draw.

How to Interpret the Results

The calculator returns an expected value, which is a long-run average over many hypothetical repetitions of the experiment under identical conditions.

• Some collections will finish sooner than the expected value.
• Others will finish later, sometimes much later, especially when n is large.
• The expected number does not give a guaranteed maximum or a deadline by which you are almost certain to finish.

For example, with n = 5 and an expectation of about 11.4 draws, it is entirely possible (though unlikely) to finish in 5 or 6 draws, and also possible to still be missing a toy after 20 draws. The expectation simply balances all these possibilities into a single average.

The spread of possible outcomes (the variance) becomes more pronounced for larger n. While this page focuses on the expected number of draws, you should keep in mind that real-world runs can vary widely around that mean.

Example Values for Small n

The table below shows the expected number of draws for a few small values of n using the exact formula E = n × H_n.

Notice how the expected number of draws grows faster than n itself. Going from 5 to 10 types does not just double the expected number of draws; it increases it by more than a factor of 2.5.

Real-World Applications

While “coupons” may sound like a toy problem, the same model appears in many domains:

• Card and sticker collecting: Estimating how many packs you need to complete a set when all cards are equally likely.
• Gacha and loot-box mechanics: Understanding how many pulls are needed, on average, to obtain each character or item at least once when there are no pity systems or rate increases.
• Software testing: Thinking of different bug classes or test cases as coupon types; the model estimates how many random tests might be needed to hit every class at least once.
• Networking and hashing: Modeling how long it takes to see all possible hash buckets or identifiers in randomly distributed traffic.
• Ecology and biology: Estimating how many samples you must take to observe every species or genetic type in a simplified, equal-probability setting.

In all of these cases, the calculator helps provide a baseline expectation assuming equal likelihood for all types. In practice, if some outcomes are rarer than others, the real expected number of draws will be even larger than this idealized model predicts.

Model Assumptions and Limitations

This calculator is based on the standard coupon collector model. To interpret results correctly, it is important to understand its assumptions and limitations:

• Independent draws: Each draw is assumed to be independent of all previous draws. The probability distribution does not change over time.
• Equal probabilities: Every coupon type is assumed to be equally likely on every draw. There are no “rare” or “ultra-rare” items in this model.
• With replacement: After each draw, the system effectively “resets,” as if you put the coupon back. You always draw from the full set of n types.
• Expected value only: The output is an average. It does not describe best-case, worst-case, or typical ranges, and it is not a guarantee.
• Variance can be large: Especially for large n, the number of draws you actually observe can fluctuate substantially around the expected value.
• Constant time per draw: If you provide Seconds Per Draw, the calculator assumes that this time is the same for every trial. In reality, your time per draw may vary.

Because of these assumptions, the calculator provides an idealized lower-bound style estimate for many practical situations. If some items are rarer than others, or if the process changes over time (for example, pity timers, rate boosts, or changing sampling protocols), then you should expect real-world completion times to be longer than the calculator suggests.

Summary and Key Takeaways

Use this tool as a way to build intuition about how long random collection processes take, to compare scenarios with different numbers of types, and to sanity-check expectations about collecting full sets in games, promotions, or experiments.