# Catching a bus: a puzzle

Suppose that the last 200 working days you have observed a neighbour walking to catch a bus, and the timings of both the neighbour and the bus are consistent with some probability distribution.

• You observe the neighbour leave his house.
When do you expect him to arrive at the bus stop?
• You observe the bus in the distance.
When do you expect it to leave the bus-stop?

Now suppose that:

• You observe the bus in the distance while the neighbour is half-way to the bus-stop, and the neighbour’s progress has been consistent with that previously observed.
When do you expect the neighbour tp get on the bus?

The puzzle is to explain the role of the previously observed probability distributions in the above estimates. A hint is provide below: ..

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If according to the established probability distributions the neighbour is almost sure to catch the bus, then it seems reasonable to apply them in this case. But if not, might the neighbour see the bus and hurry? Or might the driver wait for his regular passenger? In which case, if we have never observed neighbour being late, how can we assign probabilities?

Dave Marsay

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