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Thus, all the instances of chance that can be brought to scientific treatment and, as consequence, those which do matter must be expressed in terms of probability. These relations must assume a mathematical form, something that is offered by the resources of the calculus of probabilities. Ultimately, he postulated the principle of sufficient reason as the basis for this process, when it assumes the form of a belief in the continuity of the relations expressed by mathematical regularity:. However, in both cases, in all cases, intuition guides the process, borrowing from experience what is needed to build satisfactory rational theories.

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The idea that the notion of probability instantiates chance epistemically derives precisely from this condition. The notion of probability is the theoretical artifice that effectively brings chance to scientific fact when it allows facts to be expressed mathematically in which a given degree of predictive uncertainty is, at least temporarily, unavoidable.

Barberousse, D. Cozic, Paris: Vuibert, — Madrid , Carlos [], Le Papillon et la Tornade. Garnett, London: Macmillan and Co. II, —, Rivasseau, Progress in mathematical physics , vol. He distinguishes four kinds of hypotheses, which are actually given in two lists of three each. These are the backbone of natural science and can be seen non-controversially to be compatible with standard accounts of, say, the Hypothetical-Deductive method. They therefore may not be considered to be hypotheses at all but are however often mistaken for such. In theory though, the latter could be possible for a being whose faculties are much more sharpened than ours, but still finite.

In this case, his capability to deal with molecules should make it possible to invalidate the second law of thermodynamics [Maxwell , ]. The determinism that surrounds this argument is centered in the idea that the complete comprehension of nature is avoided due to the limits of our knowledge alone. This is why the problem deserves attention, according to Maxwell—the stability or instability of a system derives from the particular conditions through which we experience the world. See [Maxwell , ]. Plan 1 Introduction.

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He distinguishes four kinds of hypot Haut de page. A interesting example of the concerns about avoiding danger from acts of nature was the trial of a group of Italian geologists and a government official for allegedly failing to give proper warning about an earthquake that killed people in L'Aquila, Italy in Geologists around the world expressed concern that while progress had been made about trying to forecast earthquakes, there were no certainties only probabilities in the forecasts that could be made.

After a trial, seven individuals were found guilty of manslaughter and sentenced to six years in jail. In an appeals court freed the geologists and reduced the sentence of the government official to the relief of scientists around the world but citizens in the courtroom who were relatives of the those who died in L'Aquila decried that the "government" had exonerated itself. When a major storm approaches, can weather forecasters be blamed if they don't warn of the potential dangers strongly enough?

Sometimes due to forecasts, transportation systems that might be damaged by a major storm are shut down preemptively, causing great logistical and economic hardship to many people.

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If the storm doesn't materialize, as sometimes has occurred, this is frustrating to many, but the flip side, not responding strongly enough when lives might be saved, is the issue involved in what happened in Italy with regard to the earthquake forecast. Compared with weather forecasting, the forecasting of earthquakes is much less advanced. Similar issues come up every year or sometimes in multiple year cycles when certain infectious diseases e. For the young, getting an illness like whooping cough or measles can cause death. For the elderly, it is unclear if some of the vaccinations they had in their youth still are efficacious and then there is the flu, which affects the elderly in a serious way.

Whereas getting the flu for younger people is disruptive, for the elderly catching the flu can lead to pneumonia or be life threatening in other ways. So should parents vaccinate their kids and should the elderly get flu shots?

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  4. Some people get allergic reactions to vaccines but the long term history of vaccinations is to have significantly improved the life expectancy and quality of life for many. Recently there was a staggering large lottery prize of 1. One reason to have faith in the mathematics of lotteries and casino gambling is that these "businesses" thrive just as long as they have lots of customers!

    If one had a crystal ball that helped pick the right lottery ticket, one could get a lot of income. When thinking about the future either as an individual or part of a group one often has expectations for the future--sometimes good expectations and sometimes ones that seem much less appealing. One contribution of mathematics to understanding what the future might bring is the notion of expected value. When mathematicians use this term they have a very precise idea in mind but it is an idea which has many subtleties.


    One reason people are nervous about the future is that they are unsure what will happen in the future. The future seems to involve chance, randomness, stochasticness, and probability. In order to understand what is meant by expected value, we first have to say something about probability theory. We commonly hear expressions about the future similar to the following: The probability of rain is 70 percent. The chance of having another earthquake in this location is 1 in a million. What do statements of this kind mean? To answer this question one has to come back to the two pillars of mathematics--theoretical mathematics and applied mathematics.

    Theoretical mathematics builds up systems of ideas and concepts based on definitions and axioms rule systems and then deduces mathematical facts, theorems, from these constructs. Applied mathematics takes these mathematical systems and tries to use them to get insights into the world. I will try to proceed in a relatively informal manner, trying to avoid "heavy" mathematical notation and "formal" definitions.

    First of all, probability is used both in domains with finitely many outcomes and in those with infinitely many outcomes. To help build intuition I will look at the theory by using contexts to try to get the idea across. If Susan is expecting twins, there are four possibilities for the birth order of children: a boy followed by a boy, a girl followed by a girl, a boy followed by a girl, and a girl followed by a boy.

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    One might ask about simultaneous births as being other "alternatives" but we will consider only the four possibilities mentioned. In a different context we might catch a salmon going to its spawning ground in a certain river in the Western United States, and weigh this salmon.

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    The possible outcome now is a real number in a certain range of weights that salmon can display--the weight can be one of an infinite number of outcomes. From a mathematical point of view we can try to model these possibilities by imagining a set M of outcomes from some kind of "experiment" or observation of the world. The set M can be finite or it can be infinite. To each outcome m in a finite set of outcomes I will assign a real number called the probability of outcome m and I will denote that probability by P m.

    These numbers can't be assigned in a totally arbitrary way--they most obey certain properties or axioms: a. P m takes on values between 0 and 1 including 0 and 1. The sum of the values P m for all of the m in M add to 1. Note that if the probability of m occurring is P m then the probability of the complementary event, m', that m will NOT occur is P m. Note that because we only have a finite number of outcomes we don't need any idea related to limits Calculus to do the calculations. For sets M with an infinite number of outcomes we will insist that the same two conditions as above hold.

    If there were a smallest value among the numbers P m , then we could not have a sum of 1 for all the outcomes. Adding some finite number no matter how small infinitely often would result in a number bigger than 1. Thus, for infinite sets dealing with probabilities has additional subtleties. However, it is worth noting that one can find infinite sets where the probabilities of the outcomes for each individual event in the set can be nonzero.

    This can be true because there are infinite sequences of positive numbers whose sum is 1. When mathematics is used in the world we have to interpret the meaning of the mathematical constructs outside of the world of undefined terms and axioms. If one is given a collection of numbers, say the weights you have had for the last 30 days perhaps taken at the same time every morning one will see fluctuation in the numbers--they probably will not be the same. If one wants to get a sense of "pattern" for these numbers one approach is to compute some typical or "average" value.

    One very appealing such number is the mean. The mean, often called the average, is obtained by adding the numbers together and dividing by the number of measurements. The trouble with using one single number to represent a large collection of numbers is that many different kinds of data sets may have that same single number as their representative. Thus, the mean of 5, 5, 5, 5, 5, 5 is 5 while the mean of -3, -3, -3, 13, 13, 13 is also 5. One of the early developments in using numbers in science and statistics was to realize that taking the same measurement "independently" several times might be a way to get a more reliable value for the number than just measuring a quantity once.

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    8. Whenever a measurement is taken there will inevitably be some errors due to the measuring device and "human" procedure but one can try to make measurements as reliable as possible. The analogue of the mean for values that are subject to chance is a quantity called the expected value.