Binomial distribution but only failures are replaced I might be thinking about this in a completely wrong way, but I have the following problem:
Let's say we have 100 people in the population, and we know the probability of someone dying on any given day is 50%. To estimate the number of people that die on day 1, we can use a binomial distribution with 100 trials and a 50% probability of success, estimating that 50 people died. To estimate the number of deaths on day 2, we can still use a binomial distribution but this time we only have 50 trials because only 50 people remain alive (and you can't die twice).
To calculate the probability of death at the end of the 3 days, we could use:
$0.5 + (0.5*(1-0.5)) + (0.5*(1-0.5)^2) = 0.875$
My question is: is there some distribution that would help me deal with this?

At any given day $t$, the distribution of deaths in the remaining population of size $n_t$ is binomial with the probability $p$ and sample size $n_t$. There is no single distribution for all days, as it changes with time $t$.
As for probability distribution for the number of days a person in the population survives, it follows the Geometric distribution. It's a distribution of the number of "failures" till a "success" when the probability of success is constant. Here, it's the probability of dying after having lived some number of says.
