# Cox proportional hazard - can hazard be treated as instantaneous probability?

I'm doing a question on cox proportional hazard models that at the very last step leaps from hazards to probabilities, and I'm not sure how it does this. I've paraphrased it below to keep it simple, but I'm confident I've not changed any important details.

The question is what what I'd consider a fairly standard cox proportional hazard type question around how long different groups of people stay in hospital, with gender used as a single covariate. We can break up the hazard as follows:

$$\lambda(t,z) = \lambda_0(t)\exp(\beta z)$$

where $z=0$ for a male patient, and $1$ for a female. We're given the associated value of $\beta$ too, which is $0.185$.

We then get asked the following:

A male has a hazard of leaving hospital after 3 days of 0.6. Calculate the probability that a female who is still in hospital after three days is NOT discharged at that point.

Initially this seemed pretty straightforward. We can get an estimate of $\lambda_0(3)$ from the male data:

$$\lambda(3,z=0) = \lambda_0(t)\exp(0) = \lambda_0(t) = 0.6$$

and then use this to get the hazard in the female case:

$$\lambda(3,z=1) = \lambda_0(t)\exp(\beta) = 0.6 \times \exp(0.185) = 0.7219$$

Here's where I get lost though, as we need to go from a hazard to a probability at that particular instant. The answer given is as follows:

A female has a hazard of leaving of $\lambda_0(3) \exp(0.185)$. So the probability that the female is discharged is $0.7219$, and the probability she is not discharged is $1-0.7219 = 0.2781$ or 28%.

I don't see how this follows? How can we go straight from the hazard to a probability? We can relate the hazard to the ratio of the probability density function to the survivial function:

$$\lambda(3,1) = \frac{p(T=3)}{P(T>3)}$$

but I'm not seeing much else that can be obviously done with this. Possibly we can assume we're dealing with a very short time period, when $\lambda(t,1)$ can be approximated as constant during integration, and then use a taylor expansion around $e$:

$$_t p_x = \exp \left(- \int_x^{x+t} \lambda (t,1) dt \right) \\ \approx \exp \left(- \lambda(t,1)t \right) \\ \approx 1 - \lambda(t,1)t$$ but again this doesn't seem to go very far.

Is there some other standard method for going from hazards to probabilities, or common approximation? Is the phrasing of the male having a hazard of 0.6 after 3 days rather than at three days somehow significant? Otherwise I'm stuck.