Stochast-what-city?

I’m getting caught up on Radiolab and recently listened to this June 2009 podcast on Stochasticity (randomness).

As always, it is fun to hear different ways people misinterpret randomness. Paradoxical, the average person seems to simultaneously hold two seemingly divergent views of randomness. On one hand we attribute too much systematic structure to randomness, on the other we seem to find fairly unremarkable events extraordinarily random, as if caused by fate.

In the first case we often mistake long-term expected probabilities in place of random short-term results. Think of flipping a coin. In the program Deborah Nolan, a statistics professor at the University of California Berkeley, divides the radio crew into two groups, one with a real coin and another with no coin. The second group is meant to try and fool Nolan, who at this point is waiting patiently in the hall, by streaming together a random set of “heads”-“tails” chains. The other group, meanwhile, is diligently flipping an actual coin 100 times. After both pairs have recorded 100 entries they write their results on the board and call Nolan back in to see if she can distinguish the real set of coin tosses from the fake.

The results? It’s no contest. Nolan quickly and easily spots the manufactured chain. The problem is that the second group’s longest string of consecutive same-sided flips was only four entries long. The real coin toss group had a string of seven. In my view, the mistake of this first group was to substitute the expected probability of flipping a head or a tail (one-half) with the shorter-term probability of getting a long string of, say, consecutive heads. We know from common sense that there is a 50-50 shot of getting either a head or a tail for every flip. And, indeed, if we were to flip the coin an infinite number of times the number of heads would tend toward the number of tails. But this is in the long run. In the short run — that is, along the way to infinity — we would likely get very long consecutive streaks of heads and tails. It’s simply that in the long run these streaks average out. Indeed, the probability of getting a string of seven heads during a session of 100 coin flips is about one-sixth, not improbable at all. But while you’re in the middle of flipping, seven seems like forever. The group with the coin momentarily thought the coin was rigged because it just seemed so improbable that seven heads would come up in a row. We (mistakenly) think that because the expected probability of flipping a head is one-half, that any normal string of flips should not deviate greatly from our intuition: H-T-H-T-H-T.

This same phenomenon is perhaps more common in our daily lives, and more pernicious, in the form of gambling. For example, people often change their betting strategy in a game of roulette if the wheel comes up black, say, five times. Surely, the next spin is more likely to be red. But each spin is an independent event and so the wheel is no more or less likely to come up red after five spins landing on black than after 100. You are simply observing over a very, very short timeframe a roulette wheel that is essentially spinning everyday non-stop for years. That at some point during its lifespan the wheel will hit black five times in a row is not strange. What would be amazing is if that didn’t happen. But because we are human, we find significance in our being present for such a streak and ascribe meaning, mistaking what is a probabilistic fact — that there’s a 50-50 shot of the wheel landing on red — with what is happening in this moment before our eyes. We somehow think that the world is out of balance if the roulette wheel comes up black several times in a row, and that the universe is then obligated to give back what it took by letting the wheel come up red. The universe “took it” and now it “owes it back.” Unfortunately, stochasticity doesn’t work like that.

In the second case people assign unbelievably low probabilities to events that aren’t at all uncommon. Take the case of multiple lottery winners, as Radiolab does. What is the chance that a person wins a major jackpot twice? Most of us think the odds are astronomical, but they aren’t. Lots of people play lots of different lotteries all around the country, and so the chance that someone at some point will win the lottery twice is almost assured. The kicker is that if we are that person our minds seem almost designed to ascribe some sort of meaning other than pure chance. Call it fate, religion, or whatever you like.

The Radiolab segment also has a great piece about two young girls who meet by chance. I’ll save the details so I don’t spoil it for you, but trust me, once you hear the story you’ll think for sure there is some otherly-world explanation. But as Jay Koehler, a professor of finance, quickly points out we are quick to spot patterns and ascribe all sorts of meaning to events we find improbable, but are slow to pick up on all of the differences and ways that most events aren’t all that uncommon at all. I think this happens to a great extent in religion, for instance. People often look for signs and, unsurprisingly, find them in their own lives or the stories of loved ones. They assume that God is watching out for them because the events they are experiencing can’t, just can’t, be chance. In reality, any given event is likely to happen to someone at sometime, but the fact that it happens to us seems to be all we need to lend even more support to our faith.

Indeed, society would be much, much different if we all had a better understanding of stochasticity.

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