Probability of a population increasing from size $N$ to size $N + 1$ in a time interval $(t, t + dt)$?

Question

Let $\lambda$ be the birth rate and consider a time interval $(t, t + dt)$.

If we have a population of size $N$ the probability of it increasing to size $N + 1$ within the interval $(t, t + dt)$ is $\lambda * N * dt$.

But something seems wrong here as we have a probability that may be greater than $1$ which is not possible?

E.g. If $\lambda = 0.5, dt = 0.5$, then if $N$ is, say, $10$, we have that the probability of the population increasing to $11$ is $2.5$.

What am I doing wrong here?

Answer 1

If we have a population of size $N$ the probability of it increasing to size $N + 1$ within the interval $(t, t + dt)$ is $\lambda * N * dt$.

This is a loose formulation of the statement that, for every $n\geqslant0$, $$\lim_{s\to0+}\frac{P(N_{t+s}=N_t+1\mid N_t=n)}s=\lambda n.$$ So yes, as you remarked, probabilities are always in $[0,1]$, but derivatives with respect to $t$ of probabilities depending on $t$ can take any value.