If the probability of surviving $1$ month is $0.5$ and the probability of surviving $2$ months is $0.3$, I understand that the conditional probability of surviving $2$ months given having survived one month is $0.3 / 0.5 = 0.6$.
However, I'm trying to calculate the probability at the outset of dying between $1$ and $2$ months. This must mean the probability of both surviving $1$ month and surviving less than $2$ months given having survived $1$ month. How do I set up the math for this? If they were independent events, we could multiply: $0.5 \times 0.3 = 0.15$. But they're not independent, are they?
Let $A_i$ be the event of surviving at the end of month $i$. So $A_1$ is the event of surviving at the end of first month. Similarly $A_2$.
Then $A_1,A_2$ are dependent because$$P(A_1\wedge A_2)=P(\text{surviving at the end of first and second month})=P(A_2)\\\ne P(A_1)\times P(A_2)$$
In fact $A_2$ can happen only if $A_1$ does.
You want to find $P(\text{surviving at the end of first month and not surviving at the end of second month})=P(A_1\wedge A_2^C).$
First remember for any two events $P,Q,P(P|Q)+P(P^C|Q)=1$. So$$\begin{align*}&P(A_2^C|A_1)=1-\underbrace{P(A_2|A_1)}_\color{red}{\text{found by you as }0.6}=0.4\\\implies&P(A_2^C\wedge A_1)=0.4P(A_1)=0.4\times0.5=0.2.\end{align*}$$