Statistics by David J. Hand

Statistics by David J. Hand

Author:David J. Hand [Hand, David J.]
Language: eng
Format: mobi
Publisher: OUP Oxford
Published: 2008-03-05T00:00:00+00:00


Understanding probability

So that we can discuss matters of uncertainty and unpredictability without ambiguity, statistics, like any other scientific discipline, uses a precise language: the language of probability. If this is your first exposure to the language of probability, then you should be warned that, as with one’s first exposure to any new language, some effort will be required to understand it. Indeed, bearing that in mind, you might find that this chapter requires more than one reading: you might like to reread this chapter once you have reached the end of the book.

Development of the language of probability blossomed in the 17th century. Mathematicians such as Blaise Pascal, Pierre de Fermat, Christiaan Huygens, Jacob Bernoulli, and later Pierre Simon Laplace, Abraham De Moivre, Siméon-Denis Poisson, Antoine Cournot, John Venn, and others laid its foundations. By the early 20th century, all the ideas for a solid science of probability were in place, and in 1933 the Russian mathematician Andrei Kolmogorov presented a set of axioms which provided a complete formal mathematical calculus of probability. Since then, this axiom system has been almost universally adopted.

Kolmogorov’s axioms provide the machinery by which to manipulate probabilities, but they are a mathematical construction. To use this construction to make statements about the real world, it is necessary to say what the symbols in the mathematical machinery represent in that world. That is, we need to say what the mathematics ‘means’.

The probability calculus assigns numbers between 0 and 1 to uncertain events to represent the probability that they will happen. A probability of 1 means that an event is certain (e.g. the probability that, if someone looked through my study window while I was writing this book, they would have seen me seated at my desk). A probability of 0 means that an event is impossible (e.g., the probability that someone will run a marathon in ten minutes). For an event that can happen but is neither certain nor impossible, a number between 0 and 1 represents its ‘probability’ of happening.

One way of looking at this number is that it represents the degree of belief an individual has that the event will happen. Now, different people will have more or less information relating to whether the event will happen, so different people might be expected to have different degrees of belief, that is different probabilities for the event. For this reason, this view of probability is called subjective or personal probability: it depends on who is assessing the probability. It is also clear that someone’s probability might change as more information becomes available. You might start with a probability, a degree of belief, of 1/2 that a particular coin will come up heads (based on your previous experience with other tossed coins), but after observing 100 consecutive heads and no tails appear you might become suspicious and change your subjective probability that this coin will come up heads.

Tools have been developed to estimate individuals’ subjective probabilities based on betting strategies, but, as with any measurement procedure, there are practical limitations on how accurately probabilities can be estimated.



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