Larger groups have more who get at least one than smaller groups. And the larger group also will have more who get none than the smaller group.
The more tries you have, the more will be shiny. Also the more tries you have, the more will not be shiny.
If random number streams were perfectly smooth we would see this clearly. Their clumpiness makes it hard to see. Because we can isolate a small series within the stream that has more than average, and other series that have less.
The larger your group, the more likely it will be to manifest the average. The smaller your group, the more likely it will be a clump – but whether it’s a clump with more than average or a clump with fewer than average will be unpredictable. If you combine all the clumps together and they will average out.
If we had access to the code we might be able to see whether any of our actions can affect the results. Barring that, we’re left with guresses, assumptions, suspicions, and rumors. (In the final measure the surprise shiny is much more fun than the one that comes every time you expect it.)