There are $N=1000$ people.
At day $d=0$ we have $Z=1$ unkown person that is infected with a disease. Every night all of the $N$ people are sharing in groups of $2$ a room together.
At day $d=1$ we have $Z=2$ persons that are infected. At the next night every person shares a room with a new random person.
So at day $d=2$ we have $Z=4$ people who are infected. We repeat this.
What is the distribution of $Z$? What is the expected value of $Z$ in relation to $d$? Assuming nobody gets treated well, when will everybody be infected?
How do I solve this? I have troubles to find a distribution. Is $Z$ binomial distributed?
I guess $E(Z)$ is easy when knowing the distribution.
Edit: Since the restriction that nobody shares the room twice with someone made it too complicated, I removed it. If anyone still know how to do this in that case, please feel free to solve it with the restriction.