Probabilistic thinking

Most people have a vague idea about probability. We know it’s how likely or unlikely something is. For example, we might check the weather forecast and conclude that rain is likely; we know that winning the lottery is unlikely. That’s as far as most people get. This short article is going to explore probability in a little more depth and encourage you to apply it to your life.

What is probability?

If you came across probability in school, you’ll know that it’s expressed as a decimal fraction between 0 and 1. Zero represents absolutely no chance of something happening; one represents complete certainty that something will happen; 0.5 means there’s a 50/50 chance of something happening. Theoretically, nothing has a probability of 0 (impossible) or 1 (certainty). There is a small, but non-zero, chance that I created the Universe; there is a high, but not certain, chance that I will die. Such thinking is for philosophers. I’m not the Creator and I will die.

Quantifying probability is useful. Let’s go back to the weather forecast. The chance of rain on any specific day is normally expressed as a percentage. For example, the chance of rain on Monday (I’m writing this on Saturday) is 40%. In fact, the daily prediction is the highest hourly probability of rain on that day. So that 40% means that there’s one hour on Monday with a 40% chance of rain (the other times have a lower probability). Chances are, there’ll be no rain on Monday (since no hour has a probability of more than 50%).

Driving slowly and wearing a seat belt doesn’t guarantee you won’t be seriously injured in a car accident. But it helps. Every time you press the accelerator you reduce the time available to take corrective action and increase the chances of an accident. See that guy tearing through your neighbourhood in his car? Probability ensures that sooner or later his luck will run out. If you’re the unfortunate person on the receiving end, your seat-belt might save your life.

Most people don’t think probabilistically. I know this because of the number of people who participate in lotteries. You’re not going to win the lottery. Ever. Interesting side story: the Florida State Lottery was the first lottery to offer huge prizes; like most lotteries at the time, it offered lots of small (but not insubstantial) prizes until some whizz suggested combining the prizes into one huge prize — and participation went through the roof. In other words, more people bought lottery tickets when their chances of winning significantly reduced.

Is there life in the Universe?

Probabilistic thinking can answer complex questions. Take the question of intelligent life in the Universe. On the one hand, it looks unlikely. A quick look around the Solar System tells us that life (any life, even simple life) isn’t common. Looking further afield things don’t look any better. We know that life is only possible in the “Goldilocks zone” — those planets a certain distance from their local sun — otherwise the planet is too hot or too cold for life. Intelligent life takes a long time to evolve; it’s taken Humanity 4.5 billion years. It also needs exceptional environmental stability. Some cosmologists believe that the particular size and location of the Moon played a role. There are other things to consider. Maybe life came and went before it became intelligent. Maybe there was intelligent life at one time but it’s long gone. The Drake Equation attempted to put all of this together. All in all, a highly unusual set of circumstances is needed for there to be any chance of intelligent life, which might explain the Fermi Paradox. Where is everyone?

It’s not looking good. Or is it? The Universe is big. Unimaginably big. The Milky Way galaxy has 200 billion stars (suns like our Sun) and there are two trillion galaxies. Most suns have several planets. You do the maths. As Stalin noted: quantity has its own quality. No matter how vanishingly small the chances of intelligent life, the sheer size of the universe makes it highly probable. Probabilistic thinking provides an answer to the Fermi Paradox. Where is everyone? Widely dispersed across the vastness of space — so dispersed that we’ll never find them. Boring but (probably) true.

Flying

People fear flying. Once you step onto the plane, you lose control. If anything goes wrong on the flight, your chances of survival are slim. In fact, flying is the safest way to travel. The National Safety Council estimated that the average American’s chances of dying on a flight were 1 in 5,000 over their lifetime. Compare that with driving, which has a lifetime fatality rate of 1 in 85. In other words, you’re 60 times more likely to die in a car than die on a plane.

Marriage

Just under half of first-time marriages (45%) end in divorce. It might surprise you that the success rates for second and third marriages are lower (60% and 73% respectively end in divorce). But I’ve got good news if you’re one of the 6% of divorced couples who re-marry each other. Their divorce rate is 28%.

Elliott’s Law

Probabilistic thinking has a few booby-traps. One of them is working backwards from an outcome to a probability. In other words, asking “What are the chances?” after the thing has happened. We do this all the time. What are the chances of you existing (after you’re born)? What are the chances of us meeting (after we’ve met)? What are the chances of you winning the lottery (after you’ve won)? The answer is 1.0. I’ve captured this in a law:

The probability of something happening after it has happened is 100%.

The chances of you winning the lottery before the lottery is tiny; the chances of you winning the lottery after the lottery is 100%. It’s not a miracle that you exist; it’s inevitable (since you’re reading this).

Another booby-trap relates to dependent and independent events. The probability of tossing a coin and it landing “heads” is 0.5 (50%). But suppose you toss a coin three times and it lands heads each time. What are the chances of it landing heads a fourth time? The answer is still 0.5. Each coin toss is what statisticians call an “independent event” — the previous event has no connection to the next event. The “gambler’s fallacy” is an easy trap to fall into as anyone who looks for patterns in lottery numbers can attest — Lotto ball 39 was the most popular number during 2022 but it has no more, or less, chance of coming up in future than any other number (because each lottery is an independent event).

Combining probabilities

Suppose I asked: “What’s the probability of throwing four consecutive heads?” This time each coin flip is connected because you’re looking for a particular sequence of coin flips (head, head, head, head). So what are the chances?

There are two ways of combining probabilities: AND and OR. This question is an AND question since you want to know the probability of heads AND heads AND heads AND heads. AND probabilities are multiplied. The probability of throwing four consecutive heads is 0.5x0.5x0.5x05, which works out at 0.0625 or (roughly) 6%. In other words, it’s very unlikely.

The other way to combine probabilities is OR. OR probabilities are added. For example, what’s the probability of throwing a 2 or 6 with a dice? The probability of any particular number (1, 2, 3, 4, 5 or 6) is 1/6 (0.166). So the probability of throwing a 2 or a 6 is 0.166+0.166, which is 0.33 or one third.

Let’s apply this to an example. What are the chances of any two kids in a class of 20 having the same birthday? On the face of it, it looks unlikely. There are 365 days in a year and only 20 kids in that class. The probability of two specific children having the same birthday is 1/365 (0.0027). So, if you combine probabilities, you might think it would be 0.0027+0.0027+0.0027 and so on (19 times), which is 0.05 or 5%. But the question was about any two children. So you have to compare every child with every other child. There are 190 unique pairs of children in that class of 20 and each pair has a probability of 1/365, giving a probability of 190/365 or 0.52. In other words, chances are there will be a common birthday.

Relative risk

The Guardian recently reported that a plant-based diet reduces the risk of bowel cancer in men by 22%. Impressive. Maybe enough to make men of a certain age reconsider their diets. But bowel cancer only affects one in 25 people (4%). So that 22% relative reduction (let’s call it 25% for simplicity) actually reduces the absolute risk of getting bowel cancer from 4% to 3%. In other words, it would prevent one man in 100 from getting bowel cancer. You might still think that a lifetime of beans and peanuts is worth the reduced risk (it is if you’re one of the unfortunate men). Or you might not.

Understanding probability is a useful life-skill. It should encourage you to drive slowly and wear a seatbelt; it serves justice to irresponsible drivers; it tells you to save your wasted lottery money; it can help you make important life-choices. But it’s only probability. You might die in a plane crash; your third marriage might be a roaring success; a plant-based diet might save your life; we might be the only intelligent life in the Universe. But probably not.