Probability in 1000 words

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There are some words that I use a lot in this blog. This is the first in a series of attempts to “get pithy” with my explanations of these words.


Iowa Storm

Iowa storm front (photo by author)

It’s probably going to rain today. Long before there was math, the human brain figured out a “gut level” understanding of when rain was more likely than not. One way to think of mathematics is as an evolved “second language” that puts more precision into our guesses about our world.

Better precision on rainfall probability has added many multiples to the productivity of farming, and thus human survival, over the centuries. But meteorologists mean a very different thing from your likely perception when they say, “a 30% chance of rain.”

Las Vegas probabilities

Swiss mathematician Jacob Bernoulli (1654–1705) built on the work thinkers from a generation before him to formalize the probabilities of several games of chance. Bernoulli called probability “a measurable degree of certainty, necessity and chance.”

Probabilities of 2 dice

Probability of outcomes when throwing two dice. There are 6 ways to throw 7.

Probability from a gambling point of view assumes that there is a finite set of outcomes. A game with two dice, as shown above, is among the easiest to calculate. If “the house” retains at least one of those outcomes, then over time and enough throws, it knows almost exactly how much money will come their way. [1] Casinos know the probability math, and typically don’t lose money unless your name is Trump.

Some kinds of insurance get close to this near certainty for large insurers, although not for individual policyholders. I wrote recently about Medicare Part D prescription coverage as an example. Drugs covered, prices, and rates of prescribing create a near-finite set of probabilities on which to profitably base rates for any one plan.

When past is prologue

Weather is more complicated than dice. When meteorologists talk about the probability of rain, they are looking at the statistical correlation of past events and conditions. Correlation does not equal causation, but the exponential increase both in available historical data and the computer speed required to process it means that increasingly reliable correlations are now speedily mapped to current conditions around the globe.

The “big unknown” remaining in weather forecasting is the extreme sensitivity of weather to tiny changes to initial conditions, the so-called “butterfly effect” of chaos theory. [2]  The error continues to get smaller, however, through the use of Bayesian probability, which continuously feeds the forecast each day plus the actual conditions back into the model to improve the next forecast. [3]

Bayes Theorem

Think of the “rain vs. no rain” probability gap of the weather forecast as a measure of the remaining unknown causation, or “what we don’t yet know, but may someday.” [4] Our brains do tend to overestimate the accuracy of these correlation-based forecasts, however. Even if the forecast calls for “only” a 20% chance of rain, the odds of encountering rain are slightly more than rolling a 1 on a single throw of a six-sided die. Cautious bettors will carry an umbrella at those odds.

Signal and noise

Political and sports forecasts fall into yet another realm of probability. Sports data guru turned political forecaster Nate Silver wrote The Signal and the Noise in 2012. [5] His successful technique involves eliminating factors that don’t help the forecast, even when counterintuitive. These factors are “the noise” as opposed to more correlative data, which is “the signal.”

The problem with political and sports forecasts is that they often use statistical techniques designed for sampling known data sets, such as the dice probabilities above, and assume they apply to the set of voters in the next election. The natural unpredictability here likely exceeds even the “chaos theory” behind weather systems. Not only are you (and maybe they) unsure of how they will vote, but you don’t even know who will show up. A “margin of error” is often cited as a cover for this, but even that statistic is intended for known data sets. This measure, then, is really “false accuracy.”

Nate Silver’s last pre-election forecast in 2016 gave Donald Trump a 28.6% chance of winning, which many thought meant a shoo-in for Hillary Clinton. With this prediction, Trump had a greater chance than rolling either a seven or a ten in a two-dice throw. In short, pretty good odds.

And now the physicists

Most physicists and cosmologists are determinists. A comet or a meteor “with Earth’s name on it” began billions of years ago on an inviolate path that will intersect our planet sometime in the future. This object will cause major destruction to life here, although possibly millions of years from now. In this view, if there is a “probability” of this event happening or in its timing, it falls into remaining unknown causation category as noted above.

Probability does come into play, however, in quantum physics, measuring the smallest things in our universe, that of the “orbitals” of electrons and the nature of other sub-atomic particles. Physicist Sean Carroll notes:

“Quantum mechanics…comes with an entirely new set of rules, governing what happens when [sub-atomic] systems are observed or measured. Most notably, measurement outcomes cannot be predicted with perfect confidence, even in principle. The best we can do is to calculate the probability of obtaining each possible outcome.”

And yet, most of these same physicists would say that adequate determinism comes into play as the examined scale get bigger. Adequate determinism is the idea that random and probabilistic quantum events “average out” quickly with more and larger objects.

In a recent podcast conversation, physicist Max Tegmark and Sean Carroll add yet another very different concept of probability, what Tegmark calls the Type 3 Multiverse. [6] This theory says we can be “in several places at once.” The universe we perceive, in this concept, is but one instance of an infinite number of universes, and different versions of ourselves may be in more than one.  Randomness and probability, in this math, are just the perception we have when we get “cloned,” with say, 50% probability that we “flip a head” with a coin and 50% probability that we “flip a tail.” We perceive that we are on just one of those “legs,” but we may actually be on both. I still need to sleep on that.


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Notes:

  1. In the simplest two-dice form of the betting game of craps, the house pays out slightly less than the odds of the roll would indicate, keeping the rest.
  2. In the classic telling of chaos theory as defined by Edward Lorenz in 1961, the flapping of a butterfly’s wings can theoretically change the weather on the other side of the planet.
  3. For more on Bayesian statistics and how it even effects the way we think, see this earlier post.
  4. Many of the artificial intelligence-based computer weather forecasting models already “know more than we do.” In other words, they are better at predictions than our manual attempts, but we can’t always explain why the programs reached their conclusions.
  5. Silver, Nate. The Signal and the Noise: the Art and Science of Prediction. Penguin Books, 2012.
  6. Listen to Sean Carroll’s Mindscape podcast from December 2, 2019, about eleven minutes in. It will “probably” hurt your brain.

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1 thought on “Probability in 1000 words

  1. Pingback: Free Will in 1000 words – When God Plays Dice

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