Conceptualising Probability

The problem with probability is that it doesn’t really exist. Certainly it never exists in the past.

Probability is an invention we use to communicate our thoughts about how likely something is to happen. We have collectively agreed that 1 is a certain event and 0 is impossible. 0.5 means that there is just as much chance of something happening as not. We have some shared perception that 0.9 means that something is much more likely to happen than to not happen. Probability is also useful for when we want to do some calculations about something that isn’t certain. Often it is too hard to incorporate all uncertainty, so we assume certainty and put in some allowance for error.

Sometimes probability is used for things that happen over and over again, and in that case we feel we can check to see if our predication about how likely something is to happen was correct. The problem here is that we actually need things to happen a really big lot of times under the same circumstances in order to assess if we were correct. But when we are talking about the probability of a single event, that either will or won’t happen, we can’t test out if we were right or not afterwards, because by that time it either did or didn’t happen. The probability no longer exists.

Thus to say that there is a “true” probability somewhere in existence is rather contrived. The truth is that it either will happen or it won’t. The only way to know a true probability would be if this one event were to happen over and over and over, in the wonderful fiction of parallel universes. We could then count how many times it would turn out one way rather than another. At which point the universes would diverge!

However, for the interests of teaching about probability, there is the construct that there exists a “true probability” that something will happen.

Why think about probability?

What prompted these musings about probability was exploring the new NZ curriculum and companion documents, the Senior Secondary Guide and

In Level 8 (last year of secondary school) of the senior secondary guide it says, “Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability) and experimental estimates.”

And at NZC level 3 (years 5 and 6 at Primary school!) in the Key ideas in Probability it talks about “Good Model, No Model and Poor Model” This statement is referred to at all levels above level 3 as well.

I decided I needed to make sense of these two conceptual frameworks: true-model-experimental and good-poor-no, and tie it to my previous conceptual framework of classical-frequency-subjective.

Here goes!

Delicious Mandarins

Let’s make this a little more concrete with an example. We need a one-off event. What is the probability that the next mandarin I eat will be delicious? It is currently mandarin season in New Zealand, and there is nothing better than a good mandarin, with the desired combination of sweet and sour, and with plenty of juice and a good texture. But, being a natural product, there is a high level of variability in the quality of mandarins, especially when they may have parted company with the tree some time ago.

There are two possible outcomes for my future event. The mandarin will be delicious or it will not. I will decide when I eat it. Some may say that there is actually a continuum of deliciousness, but for now this is not the case. I have an internal idea of deliciousness and I will know. I think back to my previous experience with mandarins. I think about a quarter are horrible, a half are nice enough and about a quarter are delicious (using the Dr Nic scale of mandarin grading). If the mandarin I eat next belongs to the same population as the ones in my memory, then I can predict that there is a 25% probability that the mandarin will be delicious.

The NZ curriculum talks about “true” probability which implies that any value I give to the probability is only a model. It may be a model based on empirical or experimental evidence. It can be based on theoretical probabilities from vast amounts of evidence, which has given us the normal distribution. The value may be only a number dredged up from my soul, which expresses the inner feeling of how likely it is that the mandarin will be delicious, based on several decades of experience in mandarin consumption.

More examples

Let us look at some more examples:

What is the probability that:

  • I will hear a bird on the way to work?
  • the flight home will be safe?
  • it will be raining when I get to Christchurch?
  • I will get a raisin in my first spoonful of muesli?
  • I will get at least one raisin in half of my spoonfuls of muesli?
  • the shower in my hotel room will be enjoyable?
  • I will get a rare Lego ® minifigure next time I buy one?

All of these events are probabilistic and have varying degrees of certainty and varying degrees of ease of modelling.

Easy to model Hard to model
Unlikely Get a rare Lego ® minifigure Raining in Christchurch
No idea Raisin in half my spoonfuls Enjoyable shower
Likely Raisin in first spoonful Bird, safe flight home

And as I construct this table I realise also that there are varying degrees of importance. Except for the flight home, none of those examples matter. I am hoping that a safe flight home has a probability extremely close to 1. I realise that there is a possibility of an incident. And it is difficult to model. But people have modelled air safety and the universal conclusion is that it is safer than driving. So I will take the probability and fly.

Conceptual Frameworks

How do we explain the different ways that probability has been described? I will now examine the three conceptual frameworks I introduced earlier, starting with the easiest.

Traditional categorisation

This is found in some form in many elementary college statistics text books. The traditional framework has three categories –classical or “a priori”, frequency or historical, and subjective.

Classical or “a priori” – I had thought of this as being “true” probability. To me, if there are three red and three white Lego® blocks in a bag and I take one out without looking, there is a 50% chance that I will get a red one. End of story. How could it be wrong? This definition is the mathematically interesting aspect of probability. It is elegant and has cool formulas and you can make up all sorts of fun examples using it. And it is the basis of gambling.

Frequency or historical – we draw on long term results of similar trials to gain information. For example we look at the rate of germination of a certain kind of seed by experiment, and that becomes a good approximation of the likelihood that any one future seed will germinate. And it also gives us a good estimate of what proportion of seeds in the future will germinate.

Subjective – We guess! We draw on our experience of previous similar events and we take a stab at it. This is not seen as a particularly good way to come up with a probability, but when we are talking about one off events, it is impossible to assess in retrospect how good the subjective probability estimate was. There is considerable research in the field of psychology about the human ability or lack thereof to attribute subjective probabilities to events.

In teaching the three part categorisation of sources of probability I had problems with the probability of rain. Where does that fit in the three categories? It uses previous experimental data to build a model, and current data to put into the model, and then a probability is produced. I decided that there is a fourth category, that I called “modelled”. But really that isn’t correct, as they are all models.

NZ curriculum terminology

So where does this all fit in the New Zealand curriculum pronouncements about probability? There are two conceptual frameworks that are used in the document, each with three categories as follows:

True, modelled, experimental

In this framework we start with the supposition that there exists somewhere in the universe a true probability distribution. We cannot know this. Our expressions of probability are only guesses at what this might be. There are two approaches we can take to estimate this “truth”. These two approaches are not independent of each other, but often intertwined.

One is a model estimate, based on theory, such as that the probability of a single outcome is the number of equally likely ways that it can occur over the number of possible outcomes. This accounts for the probability of a red brick as opposed to a white brick, drawn at random. Another example of a modelled estimate is the use of distributions such as the binomial or normal.

In addition there is the category of experimental estimate, in which we use data to draw conclusions about what it likely to happen. This is equivalent to the frequency or historical category above. Often modelled distributions use data from an experiment also. And experimental probability relies on models as well.  The main idea is that neither the modelled nor the experimental estimate of the “true” probability distribution is the true distribution, but rather a model of some sort.

Good model, poor model, no model

The other conceptual framework stated in the NZ curriculum is that of good model, poor model and no model, which relates to fitness for purpose. When it is important to have a “correct” estimate of a probability such as for building safety, gambling machines, and life insurance, then we would put effort into getting as good a model as possible. Conversely, sometimes little effort is required. Classical models are very good models, often of trivial examples such as dice games and coin tossing. Frequency models aka experimental models may or may not be good models, depending on how many observations are included, and how much the future is similar to the past. For example, a model of sales of slide rules developed before the invention of the pocket calculator will be a poor model for current sales. The ground rules have changed. And a model built on data from five observations of is unlikely to be a good model. A poor model is not fit for purpose and requires development, unless the stakes are so low that we don’t care, or the cost of better fitting is greater than the reward.

I have problems with the concept of “no model”. I presume that is the starting point, from which we develop a model or do not develop a model if it really doesn’t matter. In my examples above I include the probability that I will hear a bird on the way to work. This is not important, but rather an idle musing. I suspect I probably will hear a bird, so long as I walk and listen. But if it rains, I may not. As I am writing this in a hotel in an unfamiliar area I have no experience on which to draw. I think this comes pretty close to “no model”. I will take a guess and say the probability is 0.8. I’m pretty sure that I will hear a bird. Of course, now that I have said this, I will listen carefully, as I would feel vindicated if I hear a bird. But if I do not hear a bird, was my estimate of the probability wrong? No – I could assume that I just happened to be in the 0.2 area of my prediction. But coming back to the “no model” concept – there is now a model. I have allocated the probability of 0.8 to the likelihood of hearing a bird. This is a model. I don’t even know if it is a good model or a poor model. I will not be walking to work this way again, so I cannot even test it out for the future, and besides, my model was only for this one day, not for all days of walking to work.

So there you have it – my totally unscholarly musings on the different categorisations of probability.

What are the implications for teaching?

We need to try not to perpetuate the idea that probability is the truth. But at the same time we do not wish to make students think that probability is without merit. Probability is a very useful, and at times highly precise way of modelling and understanding the vagaries of the universe. The more teachers can use language that implies modelling rather than rules, the better. It is common, but not strictly correct to say, “This process follows a normal distribution”. As Einstein famously and enigmatically said, “God does not play dice”. Neither does God or nature use normal distribution values to determine the outcomes of natural processes. It is better to say, “this process is usefully modelled by the normal distribution.”

We can have learning experiences that help students to appreciate certainty and uncertainty and the modelling of probabilities that are not equi-probable. Thanks to the overuse of dice and coins, it is too common for people to assess things as having equal probabilities. And students need to use experiments.  First they need to appreciate that it can take a large number of observations before we can be happy that it is a “good” model. Secondly they need to use experiments to attempt to model an otherwise unknown probability distribution. What fun can be had in such a class!

But, oh mathematical ones, do not despair – the rules are still the same, it’s just the vigour with which we state them that has changed.

Comment away!

Post Script

In case anyone is interested, here are the outcomes which now have a probability of 1, as they have already occurred.

  • I will hear a bird on the way to work? Almost the minute I walked out the door!
  • the flight home will be safe? Inasmuch as I am in one piece, it was safe.
  • it will be raining when I get to Christchurch? No it wasn’t
  • I will get a raisin in my first spoonful of muesli? I did
  • I will get at least one raisin in half of my spoonfuls of muesli? I couldn’t be bothered counting.
  • the shower in my hotel room will be enjoyable? It was okay.
  • I will get a rare Lego minifigure next time I buy one? Still in the future!

8 thoughts on “Conceptualising Probability

  1. It’s “probably” not a good idea to get too philosophical about the notion too early. Most people can grasp the concept in principle, but too much philosophizing about it can easily shake their confidence, (as indeed it should, but that can come later). One issue we can all agree upon, though, is the set of rules that probabilities must obey, if it is to have any mathematical cogency at all. In my view (and indeed my experience, when I was a teacher), time spent settling the rules of probability can be useful in demystifying the concept and can give students confidence, even if this is later undermined when the first Bayesian comes along… !

    • I fear I have to agree that talking too much about the philosophy of probability can undermine learning the tools. It’s probably good to highlight that there are some thoughtful questions to be asked, though. I had a fairly satisfying round of essay questions built around the trivia that it’s (slightly) more probable that the 13th of the month is going to be a Friday than any other day of the week, despite the fact that in the Gregorian calendar the day-of-the-week on which any day of any month falls is perfectly, precisely defined and seems not to be a matter for chance at all.

  2. i really enjoyed this post. however the examples are all very similar. i think it would have been nice to add an example from medical statistics, where say 7 sick patients were treated and 6 were cured, to then think about the probability that a new sick patient will be cured. here we can think about replications, eg by trying the treatment on more patients and thinking about how many will be cured.

  3. “But when we are talking about the probability of a single event, that either will or won’t happen, we can’t test out if we were right or not afterwards, because by that time it either did or didn’t happen. The probability no longer exists.“

    That’s not true: A time-asymmetric view of probability is tempting because we often have a very large amount of evidence that a certain event has occurred, but we can never rule out all other logical possibilities.

    “The only way to know a true probability would be if this one event were to happen over and over and over, in the wonderful fiction of parallel universes.”

    The idea of a “true probability” isn’t a good one (in my view, not even in quantum theory) but not because of the fictionality of ‘parallel’ universes: (in which you wouldn’t be able to measure it anyway!).

  4. This discussion has been very useful to me. I started musing along similar lines when confronted with the obligation to teach ‘true probability’. I think the difficulties with this concept derive from the fact that we are so accustomed to explaining probability by way of trivial examples, and the trivial examples are hopelessly inadequate in this case.
    Trivial example #1: I have cut the deck of cards, and am about to turn over the top card. What is the probability that the card will be a Heart? The ‘model estimate’ tells me 0.25. The ‘experimental estimate’ (based on data from my years of experience in a casino) tells me something very close to 0.25. What is the ‘true probability’? Clearly in this situation ‘true probability’ is totally meaningless. The card is either a heart or it is not – the only problem is that we don’t yet know which – and that is so, even before the event has happened.
    Trivial example #2: The coin is resting on my thumb, and I am about to launch it spinning into the air. What is the probability it will land showing Heads? The ‘model estimate’ tells me 0.5. The ‘experimental estimate’ (based on data from my years of experience as a conman) tells me something very close to 0.5. What is the ‘true probability’? Unlike the card example, this is not a settled question yet. But if ‘true probability’ exists in this situation, it must be a highly variable quantity, changing constantly as the muscles in my thumb flex and relax, the angle of my hand tilts, currents of air pass by, etc. The ‘true probability’ will continue to gyrate rapidly and wildly under this complex of influences, throughout the process of spinning the coin, right up to the moment when the fall of the coin is finally decided, at which point the ‘true probability’ ceases to be a probability anyway. In this case, also, ‘true probability’ if not a meaningless concept, is at least one that is of no practical use.
    On the other hand, it is not meaningless when we consider non-trivial examples.
    What is the probability that a baby will be born with a certain genetically-determined trait? We could model the situation based on known facts about the incidence of the relevant genes in the population, and get a ‘model estimate’ that way. Or we could gather some data on the past occurrence of the trait and get an ‘experimental estimate’ that way. It is not difficult to understand that both of these methods give us estimates, more or less accurate, of an unknown ‘true probability’. Is this ‘true probability’ really a probability, or is it a question that is actually decided (like the card suit) but just unknown to us? I think it is useful to think of it as a probability.

    We can act on the understanding that both our model and experimental esitmates are only estimates of a ‘true probability’ that can be approximated with increasing accuracy but never known. Einstein was wrong. God does play dice. Nature behaves in probabilistic ways, and that is why understanding the laws of probability can aid scientific investigation. True, the more we know about causal mechanisms at work in nature, the more we reduce the scope of randomness and probability. But there is no reason to believe that it can ever be eliminated.
    This is partly a matter of our historical inheritance. The science of probability arose out of the growth of both gambling and insurance (which gave us model estimates and experimental estimates respectively). For a long time, that was all that was needed. It is only more recently that the idea of a probabilistic nature has arisen, and I think the idea of ‘true probability’ is inseparably linked to that.
    How deeply we can go into this with Year 13 students is another matter again, but I think it does no harm to raise the issue.

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