What is Probability?

diceWe use phrases like "the probability of this coin coming up heads is 1/2", and "the odds on Manchester United winning their match are 2 to 1", and "the chance of dying of cancer is 30%". But what do these numbers actually mean? There are fundamentally different views about this, which can lead to very different ideas about how to deal with uncertainty.

The mathematics of probability may be sorted out, but the underlying concepts are not straightforward. Maybe that's why the study of probability only started 350 years ago, which is relatively recent compared to other mathematical ideas.

The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements about when probabilities can be used. For example, iIf I throw a die, look at the result but don't show it to you, what is the probability (for you) that it is a 6? In one interpretation it is still 1/6, since you don't know the result, but in another interpretation the probability is either 1 or 0, since the die shows either a 6, or it doesn't.

The other sections are roughly chronological:
The development of probability introduces the origins of probability theory and asks why it has only been developed so recently.


Gambling ca. 1800 (from Wikimedia Commons)

The classical approach shows the way probability has been interpreted early on, as a measure of our ignorance, in particular by the 18th - 19th century mathematician Laplace.
The logical approach interprets probability as a rational and objective degree of belief, and sees probabilistic reasoning as an extension of logical reasoning, with clear and indisputable answers.
The subjective approach , in contrast, interprets probability as a subjective degree of belief. There is no such thing as an objective probability, but instead a probability is determined by what we are willing to bet on something happening. It can be shown that, if we do not want to be vulnerable to losing money, our betting odds follow the mathematical laws of probability.
The frequency approach considers probability to be the relative frequency with which events occur in the long run: if we toss a fair coin long enough, then the relative frequency of heads against tails will tend to be 1/2.
The propensity approach considers probabilities to be inherent in the experimental set up, and was developed (at least in one of its incarnations) mainly to iron out a shortcoming with the frequency approach, namely that it does not allow us to talk of the probability of events that occur only once.

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Comments

Anonymous's picture

No mention of Kolmogorov, who from the perspective of most mathematicians defined probability in terms that encapsulates the classical, subjectivist and frequentists approaches,
Anonymous's picture

I agree with you. I found it rather odd that he wasn't mentioned amongst the people above. Can somebody clarify this for everyone's sake? Thanks. Peter L - proactol review
Anonymous's picture

The great Kolmogorov provided a rigorous mathematical structure for probability that can be used (up to some minor details) whatever one's philosophical views. Which is very useful from a mathematical perspective, but does not really address the issue of what probability actually means. Furthermore, from a subjectivist Bayesian viewpoint, K's axioms can essentially be derived from even deeper rules for rational behaviour. So I think it's Ok to treat the 'mathematicalisation' of probability as a different issue. David S
Anonymous's picture

I was in the middle of studing proabilty when I came up with the idea that flipping a coin IS NOT 1/2 or 50% or 0.5. however you want to look at it. its calcuable. If you can figure out the exact weight distribution and the exact amount of force that is put into the coin to flip, you can figure out what side it will land on. Another exaple is that if have a bord with the colours blue, green, red and orange, based on "books", the probability of it landing on blue is 1/4. its not! if you had a mechanical device that put the same amount of force in each time, it would land at the same point each. some computer simulations would show you this. Im 16 and have come up with this conclusion in 5 minutes and I havn't read anything anywhere that contradicts this! HELP PLEASE! Dan
gmp26's picture

Dan, you're right to have noticed that probabilities will change the more you know about a situation. If you know something else about a coin - it could be imperfect or have a thick edge, or be flipped in some precise mechanical way - then you might be able to better. If you know nothing else or even if you know more but the predictive calculations get to be too hard, then 50% chance of a head is a very good estimate.
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Mike Pearson

Anonymous's picture

you must have skipped over (or perhaps its not written here) where this should be a "random" and "fair" event, with each possible outcome having an equal chance of occurring. Your situation (as pointed out already) is less random because some of the parameters of the event (e.g. if you know weight distribution) are known. This means each outcome does not have equal weighting.