5.2 Probability Basics

Whenever we talk about the likelihood of some future event taking place, we talk about probability. This likelihood serves as a prediction — what we can expect to happen or not happen. For example, people might mention the odds of winning the lottery, or the probability of being hit by lightning, or to discuss the fact that it’s likelier to die in a car accident rather than an airplane crash, or to think that the odds of having a baby girl are the same as the odds of having a a baby boy. Sociologists in particular might typically be interested in an individual’s life chances, things like the probability of going to college, the probability of being unemployed, or to have a high-paying job, etc. and comparing the probabilities for any of these happening based on characteristics like race/ethnicity, gender, socioeconomic class, religion, sexual orientation, etc.

Probability is predicated on uncertainty; as the old song goes, “the future’s not ours to see”.  We use probabilities to manage the uncertainty, usually by quantifying it. For example, life expectancy at birth is the predicted longevity that a newborn will have (given current death rates). Or you might have even taken important decisions and made choices based on odds and likelihoods (i.e.. on probabilities). An entire industry — betting and gambling — is based on the fact that we don’t know what will happen but we nevertheless try to predict what might happen.

Given the dealing with uncertainty and predictions, it shouldn’t be too surprising that probability is completely and entirely theoretical. It’s an expectation for the future, which can’t be anything but abstract. (After all if something had already happened, and has become reality, we wouldn’t need to predict it or to discuss its probability of occurring.)

Let’s start with an example which is familiar to absolutely everyone, usually from an early age. At some point in your life you have likely uttered the phrase “there’s a fifty-fifty chance of…” Like “I didn’t do too well on my last test, by now there’s a fifty-fifty chance to pass the course.” Or “the traffic looks bad but it might clear up; I still have a fifty-fifty chance of making it to the job interview on time.” Or “this plan has a fifty-fifty chance of success.” Or even “these nachos look disgusting, you have a fifty-fifty chance to get food poisoning.”

A fifty-fifty chance of course means an equal probability of something happening or not. Out of two possible outcomes, either can happen with equal likelihood so it’s impossible to predict in favour of any of them.

I’m sure you know that the fifty-fifty chance expression comes from the impossibility of predicting the outcome of a flipped coin: be it heads or tails. Assuming a coin cannot possibly fall on its edge, when flipped it has only two outcomes, represented by its two sides, falling as heads or as tails. Thus, the probability of its falling on a side (a 100 percent) is divided by two — giving us 50 percent chance to get heads and 50 percent chance to get tails.

The 50/50 percent is a prediction. The moment the coin falls, one outcome has been realized and the prediction no longer applies because the event is no longer in the future.  The distinction between the factual reality (the event has happened) versus the theoretical probability^[1] (of the event happening) might seem trivially easy to make at this point but its nevertheless very important. Keep it in mind, you’ll need it for what’s to come.

Imagine you flip a coin two times in a row. Can you predict that you’ll get once heads and the other time tails? Is it possible that you get heads twice in a row? What if you flip a coin ten times? Would you get tails exactly 5 times and heads exactly 5 times? Or could you perhaps get 3 heads and 7 tails?  What about 9 times heads and 1 time tails? And what if you flip a coin a hundred times? Or more?

You might have already reasoned it, or you might have even  tried it at some point: it’s quite possible to flip a coin and get the same side twice in a row. Or three times. Or four times. Or more. (It’s even possible to flip heads ten out of ten times in a row… or even a hundred out of a hundred. In this case possible means that there is such a probability, as small as it is. Possible doesn’t mean necessarily plausible.) How do you reconcile this with the knowledge that the probability of getting heads is 50 percent?

And that — the probability — is just it. We know that theoretically with each coin toss the coin can fall as either heads or tails, and the prediction/expectation is a fifty-fifty chance.We know that in theory, if we flipped coins forever, heads and tails will average at 50 percent of the time each^[2]. We can’t flip coins forever, however, so it’s possible we get a different outcomes distribution in any finite number of times we do it (but the larger the number of times, the likelier we’ll be getting to 50/50 percent, or close^[3]).

Thus there is no contradiction in theoretically expecting a fifty-fifty chance of flipping tails out of, say, ten tosses and actually getting heads 6 times and tails only 4, as I’m sure you know. The former is a probability distribution, the latter is the observed, actual frequency distribution of the cases/observations/data. Keep this thought too.

Before we continue on to something more novel and exciting than the old coin toss example, however, let formalize our discussion a bit.