Assume a molecule that at each time step has a probability $p$ of being removed from the body. After one time step, it seems to me that these probabilities exist:
- Molecule still in body: $1 - p$
- Molecule not in body: $p$
These two probabilities nicely add up to $1$, as we would expect. After two time steps:
- Molecule still in body: $(1 - p)^2$
- Molecule not in body: $p^2$
These probabilities do not add up to $1$:
$$(1-p)^2 + p^2 = 1 - 2p + 2p^2$$
Therefore I conclude that one of those two formulae are wrong! However, intuition is not helping me here, and though I thought that this would be an easy question to google, I have not found anything helpful. Where have I gone wrong?
Should $(1-p)^2$ be replaced with $1-p^2$?
Or should $p^2$ be replaced with $1 - (1-p)^2$?
How could I have intuitively made this decision?