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pre comments on Punctuality - Arriving on Time and Math - Less Wrong Discussion
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Bell curves may be the general case, but for the non-car-owning public-transport-using among us the situation is quite different. If a train runs every 20 minutes then being 1 minute late for the train means being 20 minutes late at the destination. Being 1 minute early has no effect on the time arriving at the destination.
It makes the prep-time discontinuous I guess.
Course, in London everyone expects everyone to often be 20 minutes late coz of the damned trains, so maybe it matter less then, heh.
For this problem, you could make the distribution of the time it takes to get to the train station - you could easily compute the average time it takes for going there, and seeing that by planning to take exactly this amount of time to get there will make you 20 minutes late 50% of the time.
The prep time will only make the "late amount of time" discontinuous, it won't change the probability of being late.