The Probability Stacking Fallacy

In the world of 'wouldn't it be nice if' it is assumed that all things are possible, it's just that some things are very improbable.  Douglas Adams ridiculed this idea with the concept of the Infinite Improbability Drive, where it was merely necessary to compute the exact probability of jumping spontaneously to the desired spot in the universe in order to have it happen.


The object of this essay is to challenge the assumption that one can merely add the probabilities of a sequence of events together and thus come up with a valid probability for the whole sequence occurring.


The example I would like to use is that of a driver who always drives the same route to work, and calculates the possibility of being stopped at each of the ten sets of traffic lights that they must traverse.  They want to discover the likelihood of getting to work without being stopped at any traffic light if they attempt the journey so early that there is no other traffic.  They accept that they may never be able to experience an uninterrupted journey personally, but have a gut feeling that one day someone may have this experience.


They start their research with a measurement of the distance between traffic lights and time their cycles, from green to red and back again, and reassure themselves that it is theoretically possible to catch every individual light on green if the driver is in control of the speed at which they travel.  The difficulty comes when the driver realises that the traffic lights do not have the same periodicity as one another.  Some may take 20 seconds for a full cycle, and others take 40 seconds and some others take only 15 seconds.


Because of the differing periodicity, the timing of a certain traffic light directly affects the available window of opportunity to catch the subsequent light on green, since the driver may neither stop nor break the speed limit.  This real-world probability is smaller than the independent probability of catching just one of the lights in the sequence on green.  Because the reduced real-world probabilities are interdependent it becomes very difficult to calculate the gross possibility.  In some permutations an uninterrupted journey actually becomes impossible because the window of ‘green’ of two sequential lights is not compatible with the requirement not to stop and not to exceed the speed limit.


Many processes that are routinely ‘probability stacked’ are analysed without reference to the compatibility of their periodicity.  Since there is no practical way to compute the cascading effect of periodicity incompatibility in a real world dynamic system this results in many sequences being assigned a high level of improbability when they should in fact be evaluated as being impossible.

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