I am trying to find the probability in the following real-world inspired scenario.
If I have a finite set of whole numbers from 0 to 4 billion which I call tokens and $n$ clients. Each time a client interacts with me I give him a number from my set of tokens. So for example when the first client interacts with me I give him 0, second guy 1, third guy gets 2 and so on and so forth. When I run out of tokens I restart from 0 again. What is the probability that a client gets the same token value in two consecutive interactions.
My first thought was it should be just 1/4 billion, but I think that this may not be the case. I hope the description of the problem is not ambiguous. If so please let me know and I'll try improve it further.
Any help would be much appreciated. Thanks in advance.
PS Edit : What about $n_{C_{1}}$ times 1 out of 4 billion ?