Showing that $\lambda$ is the probability per unit time that one particle will decay in 1 second My textbook says that $\lambda$ is the probability per unit time that 1 particle will decay in one second. This makes absolutely no sense to me - I can see that it is related to probability but cannot see how it is the probability.
We have:
$$N = N_0 e^{-\lambda t} \tag A$$
Now, if the probability of a particle decaying is $p$ then we can say that at $t=1$, $N=(1-p)N_0$. Therefore:
$$1-p= e^{-\lambda}\tag B$$
Rearranging we get:
$$\lambda = \ln\biggl(\frac{1}{1-p}\biggr) \tag C$$
What is wrong with this reasoning?
EDIT:
I just want to add an example - this is what initially confused me. Say we have a probability per unit time of decay of $\frac16$. We would therefore expect the number of particles to go from $N$ to $\frac 56 N$ in one second which, using $(A)$, implies that $\frac 56 = e^{-\lambda}$ (I think this is the dodgy step?). This means that $\lambda = ln \biggl(\frac 65 \biggr) \approx 0.1823 $ and not $\frac 16 \approx 0.1667$.
 A: First, careful with units: the argument of an exponential or a logarithm must always be dimensionless.  Keeping $t$ in your (B)
$$
1-p = e^{-\lambda t} \tag{B'}
$$
and taking the logarithm of both sides gives
$$
\ln (1-p) = -\lambda t 
\tag{C'}
$$
Now we use the Taylor expansion of the natural logarithm around unity, where the function has zero value and unit slope,
$$
\ln (1 + \epsilon) \approx \epsilon \qquad\text{when }|\epsilon| \ll 1,
$$
to write
$$
-p \approx -\lambda t.
$$
This equation becomes exact in the limit of very short time intervals or very small probabilities.  You can interpret this as saying "the probability of observing a decay is proportional to how long you wait, as long as you don't wait too long," with $\lambda$ as the constant of proportionality.
In this writing $\lambda$ is the "decay probability per unit time" rather than "the probability that one particle will decay in one second"; those two statements are equivalent only when $\lambda^{-1} \gg \rm 1\,s$.  Calculus is all about infinitesimals.
With respect to your example problem: 1/6 is too big.  Try with $p=0.01$.
A: It is not a probability. It has a different name. It is also called a decay constant ($1/\lambda=\tau$, where $\tau$ is a lifetime), the state (or level) width, etc. but not probability. The right relationship is $\lambda = -\dot{p}/p$.
