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Marlon comments on Punctuality - Arriving on Time and Math - Less Wrong Discussion
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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.