If the odds of an event are one in a billion, you shouldn’t bet the farm on it. But these events actually happen all the time; they are ubiquitous. Say you’re walking along the street, and you pass a dozen parked cars. What are the odds of those cars being parked in that exact order? Assuming their drivers parked them in random spots, and assuming every parking space is filled (which is erring on the side of caution), there are a dozen factorial, or ~500 million possible configurations of cars in parking spaces. What a miracle that you saw the cars in the order you did– it was a one in 500 million shot!
Zeno’s Dartboard Paradox
Remember that trickster Zeno, who claimed you can never walk an inch, because first you’d have to walk half an inch, and then half the remainder, and so on forever? Here’s another of his knee-slappers: you can never hit a target with a dart. Proof? For sake of simplicity, let’s say the target is a dartboard with area 1. Suppose you hit the dartboard at a point p. The odds of hitting within the circle around p with area 1/2 are: at most 1/2. The probability of hitting the circle around p with area 1/4 is: at most 1/4. Hitting a circle about p with area 1/8? Yeah, you’ve got a 1/8th chance of pulling that one off, at best! This continues forever and demonstrates that the probability of hitting point p is zero. And probability zero means impossible, right?!?!
(Well, no. Probability zero events are still possible. This might seem counter-intuitive because we tend to think probabilities add up: if an event is made up of a bunch of non-overlapping smaller events, the probability of the former should be the sum of the probabilities of the latter, we intuit. And if all the constituent events have probability zero, as in the case of hitting points on Zeno’s dartboard, the sum of all those zero’s is zero. Thing is, this additive property only works when there are countably many constituent events. In Zeno’s view, the dartboard is basically a disc in the real plane, so has uncountably many points.)
The Anthropic Principle
If the value of some physical constant were off by just a tiny amount, life as we know it would be impossible. Some people argue that the fact the universe is so perfectly tuned, is itself a near impossible coincidence. The Anthropic Principle attempts to explain the coincidence: if the universe didn’t support life, then we wouldn’t be here to observe it.
Everybody who’s survived having a limb cut off, reports circumstances conducive to surviving: rapid medical attention, for example, or sheer mountaineer badassery. Based only on the available first-person anecdotal evidence, we’d be forced to conclude the chances of surviving such an injury are: 100%. But of course, that’s because dead men tell no tales. Neither do universes.
The Birthday Phenomenon
If you put 23 people in a room, there’s about a 50% chance somebody shares a birthday. Seems scandalous at first glance, but if you hash out the computations, it all makes sense. Gather 50 people, and that probability goes up to over 90%; gather 367 and a shared birthday becomes inevitable (by the Pigeonhole Principle). This isn’t an example of an ubiquitous near-impossible coincidence; just of one which intuitively seems more unlikely than it really is.
The Gambler’s Fallacy
Say you start flipping a coin, and you get four heads in a row. What are the chances the next flip will also be heads? The answer is 1/2, which shouldn’t surprise people but sometimes does anyway. The issue is that we believe five heads in a row is a pretty rare phenomenon, so after four heads in a row, a fifth seems highly unlikely! A common joke has an inept statistician (or whoever) pack a bomb before heading to the airport, reasoning: the odds of two bombs aboard the same plane are astronomically small, so bringing one bomb should guarantee a safe flight!
If the gambler’s faulty logic worked, here’s a failsafe way you could strike rich at Vegas (hat tip TheDailyWTF). Hang out where you can see all the roulette tables, and wait ’til a table shows five consecutive reds or five consecutive blacks. Then swoop in and bet on the color switching. Hey, a run of six consecutive reds or blacks is really unlikely, isn’t it, so you’re sure to win… right??
FURTHER READING
Uncountable Sums
A Provability Paradox
Right Action