(1) Suppose you have a morning commute that averages 30 minutes. However, the travel time has a high variance. Twenty percent of the time, it takes 15 minutes. Another 20 percent of the time, it takes 45 minutes. The rest of the time, your commute lasts somewhere in between. You have no way of knowing in advance.
What is your optimal behavior? Do you allow 45 minutes, even though you'll be extremely early most of the time (about four days a week)? Do you allow 30 minutes and take your chances that you won't get fired on those days (about once a week) that you'll be 15 minutes late? Or do you do something else?
(2) Suppose instead that your morning commute is usually 30 minutes, but each day there is an X% chance that it will be 45 minutes.
Certainly, you should allow 30 minutes if X is sufficiently small (say, 0.02%); you shouldn't arrive 15 minutes early every day to avoid being late once every 20 years. For larger values of X, though, the risk of being late becomes too high, compelling you to allow 45 minutes. But what is the cutoff point?
(3) Suppose instead that your morning commute is always 45 minutes, without fail.
Each day, you are spending an average of 15 minutes longer commuting (30 minutes, if you count both morning and afternoon commutes) than under the first scenario, yet your schedule is predictable from one day to the next.
Note that you could always allow 45 minutes under all three scenarios and never be late. In the first two scenarios, on those days you arrive early, you could spend some of this time at the coffee shop, or on a park bench, or getting an early start at the office. Your options are wider for that excess time, compared with the third scenario, yet you have a feeling that you're wasting a lot of time by arriving early so often.
I bet many people, myself included, would prefer the third scenario, even though it means a longer expected commute and doesn't allow you the freedom to do other things on the days you are early.
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