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Probability and the Evil Tweep of No Sleep

As I type this, the calm clicks of the keys are punctuated once a minute by a sound I can most accurately ascribe to an evil, mutant cricket. It is the smoke detector at the top of the stairs, warning me that its battery is low. It has, in fact, been warning me quite urgently and persistently since 3:22AM this morning. I do not believe the timing is random, and I suspect it is actually, motive unknown, part of a sinister plan to do me in. Having invested our full share to inflate the housing bubble, our home meets all of the recent building codes pertaining to fire safety. This includes a full complement of hard-wired smoke detectors -- one in each bedroom, at the top and bottom of any stair way and in locations within hallways whose precise specification eludes me. The net result of this is that we have no fewer than nine smoke detectors in the house. I discern at a minimum that the authors of the current building code possess significant stock holdings in smoke detector companies, even if they are not fully complicit in the threats against my life. While superficially this surfeit of protection may seem well intended, I cannot explain the low battery warnings. These devices are wired into the wall current; could not someone have designed these things with rechargeable backup batteries? Our home is three years old now and the alkalines are all predictably failing. Of the four detectors who have so far called out for assistance, each has chosen the deep, dark of night to start their cry. And this is where the inescapable logic of probability gives credence to the plot against me. With 24 hours in a day, the probability that any given smoke detector is going to reach its low battery state on a particular hour is 1/24 or roughly 0.042. We might optimistically assume my critical need for sleep occurs between 11:00PM and 5:00AM, a span of six hours. The probability of a smoke detector demanding its battery be replaced during my core sleep cycle is thus 6/24 or 0.25. A 25% chance here makes me sound like a conspiracy theorist, but I assure you this cover of reasonableness masks something much darker. If the probability that one smoke detector is going to wake me is 0.25, the odds that two consecutive detectors will both do so is 0.25 x 0.25 or 0.0625, or 1 in 16. That scenario is going to happen to a lot of people, maybe even you. The odds that three detectors will wake me are 0.25 x 0.25 x 0.25 or 0.015625, or 1 in 64. That there is just bad luck. But four detectors, 0.25 x 0.25 x 0.25 x 0.25, is 0. 0.00390625 or 1 in 256. In other words, not impossible but somewhat improbable. Even still, why do I perceive this as a threat upon my person? The clincher is that this fourth detector, the one that has so thoroughly robbed me of sleep and sanity, is located at the top of the stairway in such a position as to require an extension ladder and nerves of steel to reach. With the stairs involved, the elevation is well over 20 feet and is the highest part of the interior of the house. Any error on my part spells certain death on the unmerciful tile floor below, assuming I’m lucky enough to miss the railing. I cannot quantify this dimension of the problem, but my sleep deprived brain tells me a scenario that both demands and deprives a person of physical dexterity must be at least million-to-one odds. One in 256 odds I can stand, but the likelihood of the detector at this particular location starting its tirade at 3:22AM is simply too much to write off as chance. I am prepared to say the smoke detectors are out to get me. I’m waiting for the caffeine to fully kick in, then I will do battle with ladder and nine-volt battery in hand. If these are my last words here, you will know who struck me down.