How well does probability work in the real world?

Question

When I was first introduced to probability at the age of about 13, me and my classmates always used to ridicule the subject in a most immature manner. We used to take a deck of well shuffled cards and 'test' the results of probability. For instance the probability of picking a spade was 25%. That is a 1 in 4 chance. We tested this by picking four cards. We got 3 hearts and a dice. We didn't get the expected 1 spade card neither did we get the expected 2 black cards. Hence we decided the science was a silly one.

Looking back at those days when I'm currently majoring in math sure does make me laugh. Though I know that probability has more to it, I have not dealt with it at a theoretical level and neither do I know how to argue with children such as the ones we were once upon a time. If a child comes and claims that he tested the outcomes expected by probability and every test failed against probability's favour, I do not know how to argue with him. All I have are trust in Mathematics and the economics, sociology and quantum theory to which probability has proved to be an asset. But that is my opinion and not a proper answer.

But I want to answer the child some day....how is probability justified? What is the difference between a 50-50 chance and an indeterminable probability? Well, the 50-50 chance doesn't mean that the outcomes are going to vary alternatively (head, tail, head, tail....); the outcome is more likely (head, head, tail, tail, tail, head, tail....) doesn't my that mean it's indeterminate? If probability claims that the 50-50 chance works only in ideal situations does that mean that in such a hypothetical ideal world, the toss outcome will alternate uniformly?

I'm tempted to tell the child that the answer is that probability deals with idealised situations and that's why the results vary. But I shall be a criminal if the child assumes (because of me) that probability is simply a theoretical study of idealised situations that finds no use in real world situations. I cannot mislead the child and I cannot simply live on opinion and trust. What is the answer to the child's question?

Answer 1

There's a lot that one could say in response to your question, but I'll just make a few quick remarks.

First, it's not a result of mathematical probability theory that the probability of picking a spade is $1/4$. That's an assumption that one would make in specifying a probability model.

Why would one make such an assumption? Well, that depends on what one takes probability to be and what phenomena one is trying to model. Debates about this still rage across various disciplines including applied math, statistics, philosophy, economics, etc.

To give just one example, if we interpret probability as subjective degree of belief (for more on this, you may be interested in my answer here), one might appeal to a principle of indifference to justify the assumption that the probability of drawing a spade is $1/4$: if one has no information about the distribution of the cards, it's rational to distribute probability uniformly over all the possible outcomes. This principle is highly contentious, however. See Bertrand's paradox.

So your question "How is probability justified?" depends on many things. Moreover, some would argue that standard, mathematical (i.e. measure theoretic) probability theory is not always justified, as your question seems to presuppose. Continuing with our example of subjective probability, many prominent theorists (Savage, Dubins, etc.), following De Finetti, have argued that subjective probabilities need not be countably additive, contra the definition of a measure. Kolmogorov himself, the founder of modern mathematical probability, regarded countable additivity as a mere "expedient", useful for dealing with infinite probability spaces. See this question and my answer to it.