The Art of Indifference by Boris Ter-Avanesov (12FM)
This article will discuss the issue of multiple contradictory solutions arising in both pure and applied probability problems as a consequence of even the subtlest of variations in the interpretation of the phrase “at random”. Instrumental to this discussion are the principle of indifference, the principle of maximum ignorance and the principle of transformation groups. I explore some proposed solutions with particular focus on the work of E.T. Jaynes using Bertrand’s Paradox and other similar problems to enunciate some of the central issues in probability theory. In the physical world, we can categorise systems as being deterministic, chaotic or random. With deterministic systems, provided that we can measure the initial conditions with sufficient accuracy, we can predict the evolution of the system at any point in the future. Chaotic systems, though believed to follow deterministic physical laws, are particularly sensitive to the values of their initial conditions and whilst we might be able to predict the evolution of the system quite reliably in the short term, over longer time periods the effects of errors in the measurement of the initial conditions accumulate and give rise to unpredictable behaviour. We use these types of systems as sources of chance because they can be effectively random, such as the flip of a coin, the roll of a dice or future weather patterns. However, in truly random systems no amount of accuracy in the measurement of initial conditions equips us any the better to forecast the state of the system even at the very next instant in time. We apply the concept of chance to situations where we lack enough information to say for certain what the outcome will be, irrespective of which of the three categories the system falls into. When we have absolutely no information then as far as we are aware we are dealing with the third type of system and our best guess is to attribute equal likeliness to all possible outcomes. For this reason we use the uniform distribution or more generally the principle of indifference as the starting point in such analyses. The principle of indifference, sometimes called the principle of insufficient reason, is a very old rule of thumb that has been in use since the earliest writers on probability such as Jacob Bernoulli and Pierre Laplace. It states that if you have n options which are indistinguishable except by name then you assign equal probabilities of 1 to each of them. The principle has had success in both abstract and applied mathematics, for example with James Clerk Maxwell's predictions of the behaviour of gases. Unfortunately there are cases when it seems to lead to incorrect results. Even in deterministic physical systems we are quite prepared to see a small amount of variation in the results of experiments. We do not expect to see variation in the results of purely mathematical investigations. Surely, inconsistency in the answers to mathematical questions is seen as contradictory and hence paradoxical as we imagine these rigorous formal systems to preclude the types of subtle variation that lead to the spread of physical results. However, there are numerous examples in probability in which perfectly valid alternative methods for the same problem give rise to contradictory results.
Bertrand’s Paradox
‘Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?’ Bertrand presents three arguments to answer this problem: The ‘random endpoints method’, the ‘random radius method’, and the ‘random midpoint method’, all three of which seem valid yet yield different numerical values.
Made with FlippingBook - professional solution for displaying marketing and sales documents online