This time, we examine how different schools of mathematicians and philosophers have interpreted the concept of probability. We consider five different versions.
Classical approach
The earliest version. Thanks to people such as Laplace, Fermat, Pascal, etc, who wanted to explain the principles of games of chance (e.g., gambling). In their definition, for a random trial, the probability of outcomes equals
# of favourable cases / total # of equally possible cases.
This way, a coin has one out of two (1/2) probability to land on a head, and a dice has a one out of six (1/6) chance to land on number 4, etc.
But this leads to a problem, e.g., the probability of rain tomorrow. If the favourable outcome is rain, what are those “equally possible outcomes” – {rain, no rain}? In that case, the probability is
{rain}/{rain, no rain}
is always 1/2, which can not be true!
Logical approach
We know the format of a logical statement – premise leading to conclusions. It defines an argument as one of the two categories – deductively valid or invalid. If the premises entail the conclusion ie.e., true premises guarantee a true conclusion, it’s a valid argument. On the other hand, a conclusion which is not true, even if the premises are all true, is an example of a deductively invalid argument.
What about something in between?
Premise 1: There are 10 balls in a jar: 9 blue and 1 white
Premise 2: One ball is randomly selected
Conclusion: The selected ball is blue
The argument is deductively invalid, but we know the chance of this conclusion being right is high. In other words, the premises partially entail the conclusion. The degree of partial entitlement is probability.
Frequency approach
We all know about the frequency interpretation of probability. Take a coin, toss it and record the sequence of outcomes. Estimate the number of heads over the number of tosses. The long-term ratio or the relative frequency is the probability of heads on a coin toss.
Probability = relative frequency as the # of trials reaches infinity.
But then, what is the probability of a single-case event?
Bayesian approach
What is the probability that I pass today’s exam? Naturally, I don’t have a chance to do a hundred exams and inspect the outcomes. I must express some confidence and give a subjective (gut) feeling. In other words, the probability I assign is a degree of belief.
Note that in the Bayesian approach, we are prepared to ‘update‘ the initial degree of belief based on evidence.
Propensity approach
Propensity is a term coined by the philosopher Karl Popper. Consider the flipping of a fair coin. This philosophy school argues that it’s the physical property or the propensity of the coin that produces a head 50% of the time. And the numerical probability just represents this propensity.
Reference
Interpretations of the Probability Concept: Kevin deLaplante
