There are many reasons that I am glad that I majored in Operations Research rather than mathematics or statistics. My view of the world has been affected by the OR way of thinking, which combines hard and soft aspects. Hard aspects are the mathematics and the models, the stuff of the abstract world. Soft aspects relate to people and the reality of the concrete world. It is interesting that concrete is soft! Operations Research uses a combination of approaches to aid in decision making.
My mentor was Hans Daellenbach, who was born and grew up in Switzerland, did his postgrad study in California, and then stepped back in time several decades to make his home in Christchurch, New Zealand. Hans was ahead of his time in so many ways. The way I am writing about today was his teaching about probability and our subjective views on the likelihood of events.
Thanks to Daniel Kahneman’s publishing and 2002 Nobel prize, the work by him and Amos Tversky is reaching into the popular realm and is even in the high school mathematics curriculum, in a diluted form. Hans Daellenbach required first year students to read a paper by Tversky and Kahneman in the late 1980’s, over a decade earlier. This was not popular, either with the students or the tutors who were trying to make sense of the paper. Eventually we made up some interesting exercises in tutorials, and structured the approach enough for students to catch on. (Sometimes nearly half our students were from a non-English speaking background, so reading the paper was extremely challenging for them.) As a tutor and later a lecturer, I internalised the thinking, and it changed the way I see the world and chance.
People’s understanding of probability and chance events has an impact on how they see the world as well as the decisions they make.
For example, Kahneman introduced the idea of the availability heuristic. This means that if someone we know has been affected by a catastrophic (or wonderful) unlikely event, we will perceive the possibility of such an event as more likely. For example if someone we know has had their house broken into, then we feel less secure, as we perceive the likelihood of that as increased. Someone we know wins the lottery, and suddenly it seems possible for us. Nothing has changed in the world, but our perception has changed.
There is another easily understood concept of confirmation bias. We notice and remember events and sequences of events that reinforce or confirm our preconceived notions. “Bad things come in threes” is a wonderful example. Something bad or two things bad happen, so we look for or wait for the third, and then stop counting. Similarly we remember the times when our lucky number is lucky, and do not remember the unlucky times. We mentally record the times our hunches pay off, and quietly forget the times they don’t.
I believe in God and I believe that He guides me in my decisions in life. However I do not perceive God as a “micro-manager”. I do not believe that he has time in his day to help me to find carparks, and to send me to bargains in the supermarket. I may be wrong, and I am prepared to be proven wrong, but this is my current belief. There are many people who believe in God (or in that victim-blaming book, “The Secret”), who would disagree with me. When they see good things happen, they attribute them to the hand of God, or karma or The Secret. There are people in some cultures who do not believe in chance at all. Everything occurs as God’s will, hence the phrase, “ insha’Allah”, or “God willing”. If they are delayed in traffic, or run into a friend, or lose their job, it is because God willed it so. This is undoubtedly a simplistic explanation, but you get the idea.
Now along comes the statistics teacher and teaches probability. Mathematically there are some things for which the probability is easily modelled. Dice, cards, counters, balls in urns, socks in drawers can all have their probability modelled, using the ratio of number of chosen events over number of possible events. There are also probabilities estimated using historic frequencies, and there are subjective estimates of probabilities. Tversky and Kahnemann’s work showed how flawed humans are at the subjective estimates.
For some (most?) students probability remains “school-knowledge” and makes no difference to their behaviour and view of the world. It is easy to see this on game-shows such as “Deal or No Deal”, my autistic son’s favourite. It is clear that except for the decision to take the deal or not, there is no skill whatsoever in this game. In the Australian version, members of the audience hold the different cases and can guess what their case holds. If they get it right they win $500. When this happens they are praised – well done! When the main player is choosing cases, he or she is warned that they will need to be careful to avoid the high value cases. This is clearly impossible, as there is no way of knowing which cases contain which values. Yet they are praised, “Well done!” for cases that contain low values. Sometimes they even ask the audience members what they think they are holding in the case. This makes for entertaining television – with loud shouting at times to “Take the Deal!”. But it doesn’t imbue me with any confidence that people understand probability.
Having said that, I know that I act irrationally as well. In the 1990s there were toys called Tamagotchis which were electronic pets. To keep your pet happy you had to “play” with it, which involved guessing which way the pet would turn. I KNEW that it made NO difference which way I chose and that I would do just as well by always choosing the same direction. Yet when the pet had turned to the left four times in succession, I would choose turning to the right. Assuming a good random number generator in the pet, this was pointless. But it also didn’t matter!
So if I, who have a fairly sound understanding of probability distributions and chance, still think about which way my tamagotchi is going to turn, I suspect truly rational behaviour in the general populace with regard to probabilistic events is a long time coming! Astrologers, casinos, weather forecasters, economists, lotteries and the like will never go broke.
However there are other students for whom a better understanding of the human tendency to find patterns, and confirm beliefs could provide a challenge. Their parents may strongly believe that God intervenes often or that there is no uncertainty, only lack of knowledge. (In a way this is true, but that’s a topic for another day) Like the child who has just discovered the real source of Christmas bounty, probability models are something to ponder, and can be disturbing.
We do need to be sensitive in how we teach probability. Not only can we shake people’s beliefs, but we can also use insensitive examples. I used to joke about how car accidents are a poisson process with batching, which leads to a very irregular series. Then for the last two and a half years I have been affected by the Christchurch earthquakes. I have no sense of humour when it comes to earthquakes. None of us do. When I saw in a textbook an example of probability a building falling down as a result of an earthquake, I found that upsetting. A friend was in such a building and, though she physically survived it will be a long time before she will have a full recovery, if ever. Since then I have never used earthquakes as an example of a probabilistic event when teaching in Christchurch. I also refrain as far as possible from using other examples that may stir up pain, or try to treat them in a sober manner. Breast cancer, car accidents and tornadoes kill people and may well have affected our pupils. Just a thought.