Conflicting formulae to determine the probability that an event has occurred

Question

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?

Answer 1

The probability of being not removed from the body after 2 time steps is the probability of being not removed after 2nd time step GIVING THAT it was not already removed after the 1st time step.

So it makes sense to compute the probability of not being removed after 2nd step, only if it was not removed from the 1st step. From the whole mass of probability, which is 1, after 1st step you remain with $1-p$, which is the probability of the molecule to remain. The probability to remain in body between 1st step and 2nd step is also $1-p$, but this is from the remaining mass of probability which is $1-p$. So the final answer is: probability to remain after 2nd time step is $(1-p)^2$

[Later edit]

Alternatively, consider the probability of being out of body after 2nd time step: this is the probability of being out of body at 1st time step $p$ added with the probability of being removed at 2nd time step, giving that it was not removed at 1st step, which is $(1-p)p$. Computing the total the probaility to remain after 2nd step is $p+(1-p)p = p + p - p^2 = 2p - p^2 = 1 - 1 + 2p - p^2 = 1 - (1-p)^2$.

Now both probabilities sums to $1$, which is how it should be.

[Later edit]

The key point to note is that what happens in the 2nd time interval, depends on what happens in the 1st time interval. Those events are not independent, thus you cannot multiply the probabilities.