Confusion with relativity of simultaneity I know variations of these have probably been asked numerous times before, but I'm having trouble with this specific scenario. 
Imagine the classic Train Paradox, except instead of lighting strikes we have an observer at the centre of the train shooting laser pulses towards the rear (Event $e_1$) and front of the train (Event $e_2$). Train is moving from left to right at a relativistic velocity $v$.
For an observer on the station, the light pulse travelling towards the rear has to travel a much lesser distance since the train is moving towards it. Let this distance be $0.5-vt$. 
Obviously, station observer, who has a moving reference frame, sees the $e_1$ first. 
Let us place another man at the back of the train, since he is at rest with the train, light has to travel $0.5$ (exactly half the length of the train) to reach him. 
But according to the station observer for whom light has to travel only $0.5-vt$, the light reaches the man before it actually reaches him, in his own reference frame. How is the moving observer able to see an event before it even happened in the rest frame?
 A: The description you have provided of what happens and who perceives what is perfect. Your question is how can it happen that the platform observer (Sam) perceives that an event $A$ happened $before$ it has happened in the rest frame? 
Now, the point is that there is no set-up in which we can meaningfully talk about whether Sam has observed an event before it has happened in the rest frame or after it has happened in the rest frame. All we can talk about is which event happens first and which later in one particular frame. We can talk about whether the LASER beam hits the front first or the back first in the rest frame. We can talk about whether the LASER beam hits the front first or the back first according to the platform frame. But we can't talk whether Sam observers the beam hitting the back before or after it has happened in the rest frame. There is no defined sense to this question in our current way of describing Physics.  
A: I'll admit that I can't quite pinpoint the issues in the question... partly because I think there are some possible confusions and misconceptions in the setup. I think @Mockingbird highlighted some of the misconceptions.
So, I offer a spacetime diagram on rotated graph paper (so we can visualize the tickmarks) to help clarify the situation.
Consider a train (whose rest length $L_{0}=10$) moving to the right with velocity $v=\frac{AB}{OA}=(3/5)$, so that $\gamma=\frac{1}{\sqrt{1-v^2}}=\frac{OA}{OB}=(5/4)$.

At meeting event O [which both Station and Train assign (x=0,t=0)], light-signals are emitted.
Note: Station observes the Train to have length $L_{obs}=\frac{L_0}{\gamma}=\frac{10}{(\frac{5}{4})}=8$ (length contraction). Thus,
the Station says "the back of the train is half-a-train away: $OH=4$ units", whereas
the Train says "the back of the train is half-a-train away: $OH_0=5$ units".
(Note that Train says
$H_0=(x=-5,t=0)$ is simultaneous with event $O$,
but $H=(x=-5,t=3)$ is NOT simultaneous with event $O$ [although Station says H is simultaneous with O]  (relativity of simultaneity).)
The rear-directed signal arrives at the back of the train at event $e_1$.

*

*The Station says $e_1$ has spatial coordinate $x_{1,Station}=(-0.5L_{obs})+vt_{1,Station}$, 
where $t_{1,Station}=-\displaystyle\frac{x_{1,Station}}{c}$.
So, $x_{1,Station}(1+v)=(-0.5\frac{L_0}{\gamma})$. Thus,
$x_{1,Station}=\displaystyle\frac{(-0.5\frac{(10)}{\frac{5}{4}})}{1+\frac{3}{5}}=-2.5$ and so, $t_{1,Station}=2.5$.
Station says $e_1=(x=-2.5,t=2.5)$.


*The Train says $e_1$ has spatial coordinate $x_{1,Train}=(-0.5L_{0})$, 
where $t_{1,Train}=-\displaystyle\frac{x_{1,Train}}{c}$.
So, $x_{1,Train}=(-0.5(10))=-5$ and so $t_{1,Train}=5$.
Train says $e_1=(x=-5,t=5)$, which is 2 clock ticks after event $H$.
So, I hope this will clear up the confusion.
A: The event of the light striking the rear of the train happens once only and happens at exactly the same instant in both reference frames. The time that has elapsed since the light was emitted is less in the frame of the station that it is in the frame of the train, but that does not mean that the event happens on station before it happens on the train- it just means that the two observers allocate different values to the time coordinate for the event.
