The Quincunx and the Binomial Distribution
The quincunx (or Galton board) illustrates how in a series of random binary events, most outcomes cluster around the mean, while extreme outcomes become exponentially rarer. The board models the binomial distribution: large deviations are highly improbable in symmetric randomness.
THE IDEA
FIGURE 8: THE QUINCUNX (SIMPLIFIED)—A PINBALL MACHINE
Drop balls that, at every pin, randomly fall right or left. Above Is the most probable scenario, which greatly resembles the bell curve (a.k.a. Gaussian distribution).
Let us do another round. The third flip again doubles the number of cases, so we face eight possible outcomes. Case 1 (it was win, win in the second flip) branches out into win, win, win and win, win, lose. We add a win or lose to the end of each of the previous results. Case 2 branches out into win, lose, win and win, lose, lose. Case 3 branches out into lose, win, win and lose, win, lose. Case 4 branches out into lose, lose, win and lose, lose, lose.
We now have eight cases, all equally likely. Note that again you can group the middling outcomes where a win cancels out a loss. (In Galton's quincunx, situations where the ball falls left and then falls right, or vice versa, dominate so you end up with plenty in the middle.) The net, or cumulative, is the following: 1) three wins; 2) two wins, one loss, net one win; 3) two wins, one loss, net one win; 4) one win, two losses, net one loss; 5) two wins, one loss, net one win; 6) two losses, one win, net one loss; 7) two losses, one win, net one loss; and, finally, 8) three losses.
Out of the eight cases, the case of three wins occurs once. The case of three losses occurs once. The case of one net loss (one win, two losses) occurs three times. The case of one net win (one loss, two wins) occurs three times.
Play one more round, the fourth. There will be sixteen equally likely outcomes. You will have one case of four wins, one case of four losses, four cases of two wins, four cases of two losses, and six break-even cases.
The quincunx (its name is derived from the Latin for five) in the pinball example shows the fifth round, with thirty-two possibilities, easy to track. Such was the concept behind the quincunx used by Francis Galton. Galton was both insufficiently lazy and a bit too innocent of mathematics; instead of building the contraption, he could have worked with simpler algebra, or perhaps undertaken a thought experiment like this one.
Let's keep playing. Continue until you have forty flips. You can perform them in minutes, but we will need a calculator to work out the number of outcomes, which are taxing to our simple thought method. You will have about 1,099,511,627,776 possible combinations—more than one thousand billion. Don't bother doing the calculation manually, it is two multiplied by itself forty times, since each branch doubles at every juncture. (Recall that we added a win and a lose at the end of the alternatives of the third round to go to the fourth round, thus doubling the number of alternatives.) Of these combinations, only one will be up forty, and only one will be down forty. The rest will hover around the middle, here zero.
We can already see that in this type of randomness extremes are exceedingly rare. One in 1,099,511,627,776 is up forty out of forty tosses. If you perform the exercise of forty flips once per hour, the odds of getting 40 ups in a row are so small that it would take quite a bit of forty-flip trials to see it. Assuming you take a few breaks to eat, argue with your friends and roommates, have a beer, and sleep, you can expect to wait close to four million lifetimes to get a 40-up outcome (or a 40-down outcome) just once. And consider the following. Assume you play one additional round, for a total of 41; to get 41 straight heads would take eight million lifetimes! Going from 40 to 41 halves the odds. This is a key attribute of the nonscalable framework to analyzing randomness: extreme deviations decrease at an increasing rate. You can expect to toss 50 heads in a row once in four billion lifetimes!
When to use it
Modeling risk in financial markets: understanding that extreme events (black swans) are rare but possible, and that most of the time, returns cluster around the mean.
Quality control: predicting the number of defective items in a batch using binomial distribution, where extreme runs of defects are unlikely.
Gambling and betting: recognizing that long streaks of wins or losses are improbable, helping to avoid the gambler's fallacy.
Medical testing: calculating the probability of a certain number of positive results in a sample, given a base rate; extreme results may indicate a systemic issue rather than chance.
FROM THEORY TO EVERYDAY LIFE
In practice
Coin Flip Experiment: 40 Flips
If you flip a fair coin 40 times, there are about 1.1 trillion possible sequences. Only one sequence is all heads, and one is all tails. The chance of getting 40 heads in a row is 1 in 1,099,511,627,776. If you flip 40 times per hour, it would take nearly four million lifetimes to see this extreme outcome once. Adding just one more flip halves the odds again, demonstrating how rapidly extreme deviations become rare.
Galton's Quincunx (Pinball Machine)
The quincunx is a device where balls drop through a series of pins, bouncing left or right randomly at each pin. The balls collect in bins at the bottom, forming a bell curve. Most balls end up near the center, with very few at the extreme ends. This physical model illustrates the binomial distribution and the rarity of extreme outcomes.
A MOMENT TO REFLECT
Put it to work.
How does the quincunx demonstrate that extreme outcomes are much less likely than moderate ones? What implications does this have for interpreting rare events in real life?
Consider a real-world situation where people often overestimate the probability of extreme outcomes (e.g., winning the lottery, stock market crashes). How does the binomial distribution help correct this overestimation?
Why is it that adding just one more coin flip dramatically reduces the probability of an all-heads streak? What does this tell us about the nature of randomness and the scalability of extreme deviations?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)