To kick off the blog, here’s a translation and update of the first essay I published on my French blog in May 2017.
Let us imagine Bob.
It’s Bob’s birthday today. His friend Alice, a somewhat stingy person who also was not exactly struck with inspiration when she was selecting Bob’s present, has given him a wonderful thing: a Lotto 6/49 lottery ticket1Lotto 6/49 is perhaps the most well-known lottery game in Canada..
To play Lotto 6/49, you must pick six unique numbers from the integers 1 to 49 (included). At the time of the drawing, a computer randomly generates a set of six such numbers. You then compare your chosen numbers to the winning ones, and win a prize depending on how many of them match. (Order doesn’t matter.) Prizes go from a big nothing to about $8,811,8272All values are in Canadian dollars and come from Loto-Québec’s website as of September 2020..
Here’s a helpful table of the prizes and odds of winning3Asterisks indicate average prizes:
|Category||Prize||Odds of winning|
|6/6||$8,811,827*||1 / 13,983,816|
|5/6+B||$110,841*||1 / 2,330,636|
|5/6||$2,199*||1 / 55,492|
|4/6||$78*||1 / 1,033|
|3/6||$10||1 / 56.7|
|2/6+B||$5||1 / 81.2|
|2/6||Free play||1 / 8.3|
“X/6” means that X of your picked numbers match the computer-generated ones. “B” means the bonus number and as of writing this I have no idea what that is because I have never in fact played Lotto 6/49 and I do wonder if it should be called Lotto 7/49 then? (Additional research reveals that the bonus number is simply a 7th generated number. You win the bonus number if it matches any of your six chosen numbers. The more you know!)
Loto-Québec’s website also informs us that the odds of winning any prize, including a free play, are approximately 1 out of 6.6—that is to say, 15%.
This allows us to add the missing row to the above table:
|Category||Prize||Odds of winning|
|0/6 or 1/6
(regardless of B)
|A big nothing||5.6 / 6.6
which is about
11,865,056 / 13,983,816
Back to Bob and Alice.
Alice has picked the number 7, 10, and 20, since Bob’s birthday is September 7 and the year is 2020. She has also picked the numbers 3, 13, and 33, because three is her “lucky number.” (Alice likes to say that she’s not superstitious. As evidence of this, the number 13 is lucky to her.)
At his totally covid-safe birthday party, Bob opens the envelope and discovers the wondrous present. Let us imagine Bob’s forced smile.
Bob, ever polite, thanks Alice for the nice thought. But deep down, Bob is embarrassed. He is very much aware that lottery is an illogical and irrational thing, that buying a lottery ticket is a waste of money (or, perhaps more accurately, a voluntary tax), even as a birthday present, even if it cost a mere $3.
Bob refrains from remarking any of that. The party keeps going gleefully. The music is nice. There is beer and rhum & coke and mojitos and soon Bob is moderately drunk, which allows his opinions to exit his head without going through the filter of his politeness.
“Man, a lottery ticket. Such a shitty gift,” he says to his buddy Carlos.
Unbeknownst to him, Alice was within reach of his alcohol-amplified voice. A discussion ensues. The topics under consideration include: “It’s the thought that counts,” “True, you probably won’t win anything, but it’s fun to dream about what you could do with the money,” and “You have a chance. It’s not zero.”
The first two quotes are somewhat problematic. But it’s the third one that makes Bob flinch.
Alice keeps going. “There’s a huge difference between 0 out of 14 million and 1 out of 14 million,” she says. A few other friends nod.
But Bob gets angry. “No,” he says loudly, “the difference between 0 and 1 out of 14 million is exactly 1 out of 14 million, and that number is in practice nearly zero.”
“Nearly!” Alice exclaims triumphantly.
Bob insists. He explains that “nearly zero” is just a probabilistic way to say “zero,” because a true rational mind will never consider anything to be exactly 0% likely, or, for that matter, 100% likely, since no one can ever be perfectly certain of anything—
Bob stops, realizing he’s just shot himself in the foot.
“Exactly what I was saying!” Alice says. “We’re never perfectly certain of anything, so you can’t be certain you won’t win.”
Bob sighs. Three beers and two cocktails are cycling in his circulatory system. He isn’t in full possession of his faculties. He needs help. Let’s step in.
Bob and Alice suffer from cognitive biases
They’re not alone.
The human brain, an amazingly complex and capable thing, is also poorly designed. Cognitive biases are natural tendencies to think and act in ways that don’t match reality or don’t allow us to achieve our goals. For instance, according higher credibility to a statement that supports are preexisting beliefs (confirmation bias) or finishing your plate at a restaurant even if you’re not hungry, since it’s already paid for” (sunk cost).
Cognitive biases are all over the place. When we talk about probabilities, they become especially salient. One well-known example is the gambling fallacy, in which you’re tempted to bet on “tails” after having seen three “heads” in a row, since you feel the universe must rebalance the results. Or, going back to Lotto 6/49, studying the winning numbers in past drawings so you can bet on those that have been drawn less often.
What makes Alice say there’s a “huge difference” between 0 out of 14 million and 1 out of 14 million is another bias.
Eliezer Yudkowsky writes that even if you can describe, using mathematics, such tiny numbers as 1 / 14,000,000, or even 1 / 1 googolplex4One googol = 10100 = 1 followed with 100 zeroes = 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000. The number of atoms in the Universe is much smaller than one googol. In fact it is about a hundred billion billion times smaller. One googolplex = 10googol = 1 followed with googol zeroes., the human brain isn’t equipped to feel numbers of these sizes. If you calculate an infinitesimal probability, your brain understands that this probability has some importance, since you had to spend some effort calculating it (or reading it in a table on this blog). Even the lowest imaginable perception level is too intense for what a number this small actually represents.
Let’s illustrate that with a grayscale analogy.
A grayscale analogy
Here’s a shade of gray (or brightness) of zero, according to the hexadecimal system used on the web (#0000005The hexadecimal system— which uses 16 digits, 0 1 2 3 4 5 6 7 8 9 A B C D E F—represents colors as combinations of three values, red, green and blue. Each can be anywhere from zero (00) to 255 (FF). So #FF00CC represents a combination of 255 red, 0 green and 204 blue (CC in hexadecimal = 204 in decimal) which yields this elegant hot pink. When the three values are equal, we get a color on the grayscale.).
Let’s say this represents a probability of zero for some given event. At the other end of the scale, we get a level of 100% (#FFFFFF).
This color represents the certainty that the event will occur.
Now this is what a probability of 50% looks like. A perfectly balanced coin would be a real-world example (#080808).
You can see where I’m going. A probability of 1 out of 14 million will be very close to pitch black. How much?
In hexadecimal, this color is called #010101. In other words, the values for red, green, and blue are equal to 1 out of 255 (it would be 0 for perfect blackness). Let’s compare #000000 (left) and #010101 (right):
See the difference?
I don’t. I can look at my screen from any possible angle and my poor human eyes will only see a single color. I need to increase the right-hand side’s grayscale value to #060606 to be able to perceive a difference, and even then, I can’t guarantee it’s not the result of my imagination.
If we increase the difference further, it eventually becomes apparent. But let’s go back to #010101. On a scale where 0 is absolute black, and 255 is absolute white, we have 254 shades of gray. #010101 is the darkest one. In the probability analogy, #010101 represents odds of 1 out of 255.
1 out of 255 is much, much greater than 1 out of 13,983,816. How much greater? 54,838 times greater.
Try to imagine a gray that is 54,838 times darker than #010101.
Try to imagine a gray that corresponds to 1 out of a billion. 1 out of a trillion. 1 out of a googolplex.
All these shades of gray are impossible to distinguish from absolute black. And all these probabilities are impossible to distinguish from zero. In fact, if we leave mathematical truths behind and examine reality, these infinitesimal probabilities are as close to zero as you can get. Like Bob was trying to say earlier, in real life, there never are actual 0% or 100% probabilities, because you should always allow a tiny chance that the facts contradict whatever your most certain beliefs are. The odds that the sun rises tomorrow are nearly 100%—but there’s a vanishingly small chance that we’re in a simulated universe whose creator suddenly decided to vaporize the sun during the night. The odds of that happening are nearly zero. In practical—but not absolute—terms, they are zero.
In practical—but not absolute—terms, the odds of winning the Lotto 6/49 are also zero. (At least, the odds of winning the big prize. The odds of winning anything, most likely a free play, are of 1 out of 6.6, which is approximately gray #272727—which is still much too dark for me6In case you didn’t notice, the color’s number is written in that color..)
And to reply to Alice (“we’re never perfectly certain of anything, so you can’t be certain you won’t win”), I call upon Yudkowsky again7I guess you could say this blog post is a rewrite of some of his ideas to emphasize the cognitive error he calls “the fallacy of gray.” Everything is not white or black, but that doesn’t mean all shades of gray are the same. In the same way, nothing is true or false with absolute certainty—but you shouldn’t conclude that different shades of uncertainty carry no meaningful information.
Your analogy fails because of X
Of course, this analogy is imperfect. As are all analogies.
The 256 shades of gray that can be seen on the web are but an approximation of the infinite spectrum of probabilities from 0% to 100%. And the spectrum of different shades a human eye can detect (about thirty8according to this random article) is an even less precise approximation.
Besides, 1 out of 255, even if very small, may not be negligible. At what point does a probability become so small you can discount it entirely? I don’t think there’s a universal answer to this question.
Let’s add that a linear scale is not necessarily the best way to represent probabilities. A logarithmic scale would be more useful. This may also be true of human grayscale perception.
And yet I think the analogy has value. The human brain cannot adequately feel probabilities. A sensory representation might help us grasp them better and help us take better decisions.
Which brings us to the final point of this post.
The grayscale method is not useful merely for games or for other explicitly probabilistic contexts—such as opinion polls. In fact, it applies to anything.
What is the shade of gray that the sun rises tomorrow? Based on my previous experience, I say: quasi white (#fefefe).
What is the shade of gray that the stock of [INSERT COMPANY IN WHICH YOU HAVE TOO MUCH STOCK] rise during the week? Maybe #c6c6c6?
What is the shade of gray that I’m going to enjoy vacuuming later today? Quite dark gray. But the shade of gray that I’ll enjoy living in a clean place is closer to white.
What is the shade of gray that God exists? Very dark gray, for me, but maybe not nearly absolute black. For others, it’ll be very pale gray or quasi-white (remember, absolute certainty can never exist).
The grayscale method can be generalized to anything you know (or think you know). It allows you to systematize your degree of confidence about each belief you have. Which is important, if you want your knowledge of the world to be precise, and your cognitive biases to be thwarted.
Of course, the underlying universal idea is simply probability. But probability is math, and in most cases, human brains aren’t big fans of math. So if, like Alice, you don’t love probability, you can at least try to see life in gray.
Grayscale skyline of Montreal as it appeared from where I was living in 2017
A few endnotes
- It’s so annoying that a basic concept like shades of gray always makes you think of erotic BDSM romance because of E.L. James.
- Researching web color representations made me realize I don’t know shit about color, and that it’s a much deeper topic than commonly thought9This is still as true in 2020 as it was in 2017.. (Also, it’s possible that #000000 and #010101 are really identical depending on your display. I don’t really know.) It would be worthwhile to explore this topic in another post.
- Alice and Bob have made peace, if you were wondering about the life of fictional characters. Bob didn’t win anything with his 6/49, though. Not even a free play.