Luck — a mathematical discussion

topLuck is something a lot of people care about and they often go to great lengths to call up or control luck. Mathematically, luck can be explained by random chance — which is beyond control and which is very even-handed in how it bestows good or bad luck. Superstitions are pervasive in human societies. Different societies — and different people — have firmly held beliefs about what is lucky or unlucky. The mathematical science of statistics has, at its core, the quantification of chance. Statistics takes us part of the way to replacing the notion of luck with the notion of chance.

There is an important point to this. Chance is even-handed and brutally fair. Luck is often seen as being the result of virtue. People whom chance favors are often called lucky (especially by themselves) and people who experience bad outcomes by chance are called unlucky. The search for ways to summon up, invoke, and control luck is a huge human preoccupation, from a refusal to wear clean socks in major league baseball teams to the belief in auspicious and inauspicious numbers like seven and thirteen in European cultures. A revealing fact is that the “good” and “bad” numbers vary from culture to culture.

“Luck” through reason and care


You can bend the odds in your favor, and Occupy Math urges you to do so. This involves a lot of non-luck-based practices, like looking both ways before you cross the street, eating a healthy diet (healthy for your unique needs — this is a huge issue that might make a good post), and avoiding unnecessary dangers like taking horse dewormer to fight a virus when there is no evidence it helps. One reason Occupy Math pushes the idea that math is the right of all free people is that even a little bit of math can give you the tools to find these “lucky” choices that let you avoid both obvious and subtle pitfalls.

It is possible to bend the odds against bad things happening a great deal more through the action of whole societies. Countries with universal healthcare have better health outcomes, on average, than countries that lack such care. Carefully designed traffic regulations and the layout of streets and signs can make both driver and pedestrian much safer. Educating yourself so that you are more likely to notice that someone is lying or deceived when they give you advice works much better when society has created an effective educational system. Occupy Math swears that this is elementary — an individual can learn and play the odds in many ways, but society can do the same to much greater effect.

The belief that luck measures virtue


Superstition can occur within any faith. American Christianity has found a particularly pernicious superstition, the “prosperity gospel”. A version of the prosperity gospel tells its adherents that faith, advocacy for the correct religion, and donations to religious causes will result in health and wealth as a reward from the Almighty. This belief is a weird fusion of capitalism and Protestantism that is inconsistent with a great deal of Christian theology — but it’s also a wonderful example of people finding a reason to believe that good luck is the result of virtue. The support for the prosperity gospel lies in bad sampling. All cases that support the doctrine are included in testimonials while those that contradict the doctrine are ignored. Any time you hear about a way of invoking the forces that run the universe to your personal benefit, it’s worth thinking about how the evidence supporting the belief was sampled. It is extremely common for proponents of an idea to filter their samples, their testimonials, and their examples so that they report only the positive outcomes.

Being able to notice cause and effect relationships is enormously useful. It is the keystone of adopting good practices and effective behaviors. As with all good things, it can be overdone. Human beings have a hair-trigger cause and effect detector that leads not only to good outcomes, but to magical thinking. Magical thinking is the belief a cause and effect relationship is present even though the mechanism itself is not apparent or is completely implausible. The connection between wearing dirty socks and winning a baseball game, for example, may be real — but it rests in the minds of the players. Sometimes magical thinking leads to the discovery of a new mechanism, but usually it is just wrong. It often arises in situations where a person or situation is desperate and really wants a way out of an impossible situation.

Teaching about luck


One of the most popular Occupy Math posts is Three probability puzzles that will fool you for sure. The second puzzle involves flipping a coin twenty times and then reporting the longest run of heads or of tails. If one person does this, the results are usually unimpressive. If you have a whole classroom do it (or do it fifty times), then the results become a bit uncanny. The last time Occupy Math did this, one student got eleven heads in a row and protested that “my coin is broken”.

The longest run in a sequence of coin flips is something that has pretty high variability and so it is a good tool for teaching about luck. The full activity has each student write down twenty heads and tails and find the longest run “as they think the coin will behave” and then contrast this with the actual outcomes of flipping coins. People have very bad intuition about randomness and their sense is that long runs are impossible, which helps highlight the contrast between the written down and real coin flips.

Why taking credit for your luck can be harmful


A person’s success is partly the result of their hard work and cleverness and partly the result of random chance. Successful people often over-emphasize their cleverness and effort and minimize the impact of their luck. Where a peacock would hold up his tail, a human being boasts about their cleverness. This, in turn, causes people to emulate the successful person in an attempt to succeed. This is problematic for a number of reasons.

First of all, following the same path does not often result in the same outcome because of the way the random chance parts break. Second, the first person to follow a path does not have someone on the path ahead of them. Founding Amazon, for example, is not possible for a second person — Amazon already exists. Keep also firmly in mind that you do not hear about the failures — the sampling behind success stories is incredibly biased.

If you keep these things in mind, then there is substantial potential gain in learning from the cleverness of others (this enhances your cleverness) and hard work is a part of many success stories. It also helps if you understand the role of chance, so that you can hedge your bets against unfortunate chances as much as possible, and so that you have fallback positions in case chance does not break your way. To put it another way, avoid magical thinking by understanding as much as you can about the connections between actions and outcomes. This does not guarantee success, but it reduces the odds of failure.

Mathematics advances through cleverness. After you get the basics in place, you figure out more math by having one or more clever ideas that you then have to convince other mathematicians are also correct. Often the other mathematicians catch an error and your idea gets the all-too-common badge of “clever, but wrong”. Occupy Math mentions this by way of emphasizing that, while you should keep random chance always in mind, studying the cleverness of your fellow human beings is a gold mine.

I hope to see you here again,
So remember to get your Covid vaccination!
Daniel Ashlock,
University of Guelph
Department of Mathematics and Statistics


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