Sometimes we forget to question things that seem so obvious to us that we are blind to them. Many years ago when I first flew to Europe, the airport had a sign for rest rooms. I looked for men and women logos, but quickly discovered the rest rooms were little sleeping areas that could be rented for a few hours between flights. Yeah, room to rest in. Rest rooms. Got it.
Similarly consider the word toadstools. We all know what a toadstool is, but does it derive from toad stool?
So what about probability? We use the word in common language without thinking what it means. And in fact, we often mean different things. How do we relate the probability of a dice throw coming up seven to the probability of rain tomorrow? Is it probable you know what you are talking about? What is the probability of having breast cancer? Is that a well-defined concept?
We have no problem with the dice issue. The probability that any number will come up can be calculated by simply assuming each side has an equally likely change of coming up and that the two events (that is, the two tossed dice) are independent. However, when you ask most people what it means to say that we have a 30% chance of rain tomorrow, they will not be able to give an operational definition of what that means. But even though they might not be able to define the probability, they know that 30% is more likely than 20% and less likely than 40%. But if it is not raining today and the weatherman predicts rain tomorrow with 100% probability, what does that mean?
Part of the difficulty in nailing down real world probabilities is that they change dramatically with increasing knowledge. If I ask you to estimate the probability that a person has breast cancer, and later tell you that person is a woman, your estimate should change. If I then add that she is over fifty and in good health, it should change again. Then if I tell you her mother, grandmother, and both aunts had it, what is your response? Finally, I tell you she just had a mammogram and it came back negative. Is there a way to make sense of all these gyrations?
Yes, and there are many good books that develop the whole theory of multi-variant probabilities, but that is not the issue here. In this necessarily short article, I only want to make the single point that we need to be careful of the working definitions we use, and that we have to be particularly careful of those slippery words we use in common language.
When I taught introductory physics, we counted up the new, or more rigorous definitions that were required of our students and found that the vocabulary we expected them to learn was similar in size to learning a new conversational language. For instance, I had great difficulty convincing students that speed and velocity were not the same thing.
In the same way, as we look through the decision theory-oriented puzzles and problems, it’s good now and then to step back and make sure we all agree on what we are discussing.