Supose you have two baskets with one ball each. Probability that one basket has k ball in 0-th time is 

\eqn{p_{0,k}} 

therefore 

\eqn{p_{0,1} = 1} 

and for k != 1 

\eqn{p_{0,k} = 0}.

Now (You) choose random ball from any basket (every ball have the same probability of being chosen, not dependent on which basket is it placed in). You put new ball in a basket where this choosen ball was. :0 You repeat this t times. 

In math language it is probably: 

\eqn{p_{t,k} = p_{t-1,k-1}\frac{k-1}{t+1} + p_{t-1,k}\frac{t-k}{t+1}}

Now

The most bone-chilling, slow-burn, atmosphere-oozing thing I discovered about it:

\eqn{p_{t,k} = \frac{1}{t+1}}

HOLY SCIENCE!!!!!!!!1

Probability of having k balls after t time is always the same!!!!!