One of the differences between modern times and previous ones is that we have developed the idea of numerical probability. Some authors have wondered that as smart as the Greeks were and as much as they cared about mathematics, why they didn't invent probability. The modern concept of probability is often said to have begun with correspondence between Pascal and Fermat in the mid-1600's. They were asked by a gambler which of two strategies was really better and why. The difference was very slight so the gentleman must have gambled quite a bit to be able to sense that there seemed to be a difference.
In their usual quiet way, the Italians had started writing on the topic well before these two Frenchmen but Cardano, the gambling scholar, didn't get much credit until recently.
For a long time, the subject was somewhat undeveloped and not considered a genuine branch of mathematics. It helped thinkers immensely when, in the 1900's, Kolmolgorov developed three simple axioms to base probability on:
A probability of an event is a NUMBER equal to 0 or 1 or any real number between those limits
The probability of a certainty is 1 and that of an impossibility is 0
The probability of the occurrence of an event plus the probability of the non-occurrence of that event = 1
In my work with students, I have found that the first step is to repeatedly stress that a probability is a number. I ask a student to state a probability. Someone who does not grasp that a probability is a number will , instead, state an event. Such a student may say something like "it will rain today", a statement of an event, not a number. I am trying to get them to state a number, such as 1/2 or .37, a number which could be the probability of rain today.
Because in real life, events either happen or they do not, it seems wrong to use a sliding scale to represent how likely something is. It is not unusual for people to say that they want to know whether it is going to rain, not some number that still allows for rain and for not rain. I have some sympathy, but I see that in the most important matter money people use use to a sliding scale. We are accustomed to seeing how much money we have in an account and then gauging whether we should buy a given item. We are ok with using the sliding scale of the account balance to decide if we can afford something. We don't ask,"Well, do I have enough money or not?"
Similarly, if the probability of rain is 30%, the rain likelihood balance is low but we might possibly scrape by and get some. Just don't count on it. The odds are 7 to 3 (70% vs. 30%) against it, sometimes written "7:3 against ."
The most fundamental change in probability since its invention, it seems to me, is the development of Bayesian/subjective probability and statistics. That spreading discipline can make use of personal judgments expressed numerically. The method is much like a marketplace where I get to say what I will sell my cap for. The selling price I set is my judgment of how much I want for the hat and how much people are willing to pay. Similarly, a Bayesian analysis can use numerical judgments expressed in probabilities. How likely is it that it will rain today? How likely that a given piece of legislation will be passed by the state legislature? Your estimate or mine can be used to start a Bayesian analysis of the problem going. As we gather evidence and evaluate it numerically, our answers usually converge toward a given value.
