What I did:
I found the Cumulative distribution, which is:
$$P(X\leq x)=1-e^{-\frac{x}{\theta}}$$
Then
I know that :
$$P(X>a+b|X>a) \cdot P(X>b) = P(X>a|X>a+b) \cdot P(X>a+b)$$
So we have to find $P(X>a+b|X>a)$ , so therefore I re-arranged:
$$P(X>a+b|X>a) = \frac{P(X>a|X>a+b) \cdot P(X>a+b)}{P(X>b)}$$
So I found whatever I know to find :
$P(X>a+b)=e^{\frac{-(a+b)}{\theta}}$
$P(X>b)=e^{\frac{-b}{\theta}}$
But I dont know how to find :
$$P(X>a|X>a+b)$$
Please help me how to calculate the value.
Also, Assumming this Identity. I tried to think about the next part, I don't know the meaning of this equation when X is the number of days. Please shed some light.
I know that
$P(X>a|X>a+b)=P(X>b)$
Kindoff Means:
The probability of the number of days " X" is greater than b is the same as the probability of the number of days being greater than a+b , given that the number of days is greater than b.
How do I write this in a meaningful way concerning treatment?
Since you have calculated the cmf you are almost done, since $$P(X>a+b|X>b)=\frac{P(X>a+b \text{ and } X>b)}{P(X>b)}=\frac{P(X>a+b)}{P(X>b)}=\frac{e^{\frac{-a+b}{\theta}}}{e^{\frac{-b}{\theta}}}=e^{\frac{-a}{\theta}}$$
The meaning of the equation is the memoryless property of the exponential distribution. Whether you are at day say $0$ or at day $b$ then the probability that patient needs $a$ more days of treatment is the same.