| Nope. Let's take a small example. We have 20 bins. You are going to put 3 things in those bins, with each thing getting its own bin. I'm going to roll a d20, which gives me a uniformly distributed integer from 1 through 20. I'll look into the bin with that number and if there is something there take it. You don't want me to take any of your things. How can you distribute your 3 items among those 20 bins to minimize the chances that I will take one of your items? If I were not using a uniformly distributed integer from 1 through 20 and you knew that and knew something about the distribution you could pick bins that my loaded d20 is less likely to choose. But since my d20 is not loaded, each bin has a 1/20 chance of being the one I try to steal from. Your placement of an item has no affect on the probability that I will get it. It works the same the other way around. If you place the items using a uniform distribution, then it doesn't matter if I use a loaded d20, or even just always pick bin 1. I'll have the same chance of getting an item no matter how I generate my pick. In general when you have two parties each picking a number from a given space, if one of the parties is picking uniformly then nothing the other party can do affects the probability of both picking the same number. |
Now imagine a number of these bins _can hold zero things_ (not 3). Eg in a world where all bins are the same size, you can always steal 3 things from any of the bins, whereas in a world where the bin sizes vary. You'd hit a few bins which are guaranteed empty. Doesn't this directly affect the probabilities?