Bengt is walking along an infinite straight road. He has 1000 s left to walk. But he also needs to cross the road. The safest way to cross is of course to go straight across, perpendicular to the road, which would take another 10 s. But Bengt is determined to cross the road the fastest way possible. What makes it more complicated is that there are cars arriving at random. The average frequency of cars is one per 100 s, and he can spot them 10 s in advance. To a good approximation, the cars are moving infinitely fast (so Bengt's movement does not affect the time it takes the car to reach him), they are very wide (so Bengt has to be completely across before any car reaches him) and there is no correlation between their arrivals.
So he wants to figure out the right path to follow in order to minimise the expectation value of the time. As soon as he sees a car, the answer is obviously a straight line from his current location, such that he reaches the other side in 10 s. At the moment he can't see any cars, but they could appear at any time.
Which path should he set out to take across the road?
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