Why isn't the probability of an event happening in $L$ days given the probability $p$ it happens on any day $\frac{(2^L - 1)}{p^{-L}}$?

Question

I feel like this is probably a dumb question, but there is something I am fundamentally misunderstanding here.

The problem is essentially the same as the one mentioned in this question. I understand the reasoning behind the accepted answer. If we have the probability $p$ of an event occurring on any given day, then it makes sense that the probability of the event occurring at least once over $L$ days is the same as $1$ minus the probability that it doesnt happen on every day, that is $1 - (1 - p)^L$.

However, I tried to solve this problem in a different way, and came up with a different, and completely ridiculous answer, and so there must be some fundamental flaw in my reasoning that I am not understanding.

The reasoning goes like this: on any given day the probability $p$ of the event occurring can be thought of as the probability of selecting 1 out of $\frac{1}{p}$ symbols. Thus, over $L$ days, we can think of the possibility space as the number of the number of $\frac{1}{p}$-ary strings, which should be $\frac{1}{p}^L$.

I am thinking the number of ways in which the event happens on at least one of the days should be

$$\sum_{i=1}^L {L \choose i} = 2^L -1$$

since we are 'choosing' $i$ days for the events to happen.

Thus, the probability should be the number of possibilities where it happens over the total number of possibilities

$$\frac{2^L - 1}{p^{-L}}$$

This is obviously wrong. The main reason it is wrong is that the probability of the event occurring decreases, rather than increases, with $L$. I have a feeling that this argument is really dumb, but I cant figure out why.

The main thing I am looking for is a 'constructive' reason why $1 - (1 - p)^L$ is the correct probability. If we think of probability as a form of logic, the correct reasoning, which seems to me to conflate the double negation of the truth of a proposition with the truth of a proposition, feels non constructive in nature. I don't really have a problem with this, but would just like to see a constructive reason to help me in understanding.

Answer 1

Assuming $1/p$ integer for the ease of mind, the correct numerator should be $$N=\sum_{i=1}^L {L\choose i}(1/p-1)^{L-i}$$ because you're only counting the days which the event happens, but not the individual chosen objects on the days the event doesn't happen, i.e. among $1/p-1$ ways of choosing them.

We can manipulate this equation and use the binomial expansion formula to come up with a compact version:

$$\begin{align}N&=\left(\sum_{i=0}^L {L\choose i}(1/p-1)^{L-i}1^i\right)-(1/p-1)^L=(1/p-1+1)^L-(1/p-1)^L\\&=1/p^L-(1/p-1)^L\end{align}$$

If you divide this by $p^{-L}$, it'll be $1-(1-p)^L$.