Outline an argument that may have led Pascal to this conclusion and critically consider one objection to it.
Early probability Games of chance The modern mathematics of chance is usually dated to a correspondence between the French mathematicians Pierre de Fermat and Blaise Pascal in Suppose two players, A and B, are playing a three-point game, each having wagered 32 pistoles, and are interrupted after A has two points and B has one.
How much should each receive? Fermat and Pascal proposed somewhat different solutions, though they agreed about the numerical answer. Each undertook to define a set of equal or symmetrical cases, then to answer the problem by comparing the number for A with that for B.
Fermat, however, gave his answer in terms of the chances, or probabilities. He reasoned that two more games would suffice in any case to determine a victory.
There are four possible outcomes, each equally likely in a fair game of chance. Of these four sequences, only the last would result in a victory for B.
Thus, the odds for A are 3: In that case, the positions of A and B would be equal, each having won two games, and each would be entitled to 32 pistoles.
A should receive his portion in any case. This first round can now be treated as a fair game for this stake of 32 pistoles, so that each player has an expectation of Games of chance such as this one provided model problems for the theory of chances during its early period, and indeed they remain staples of the textbooks.
Fermat and Pascal were not the first to give mathematical solutions to problems such as these. More than a century earlier, the Italian mathematician, physician, and gambler Girolamo Cardano calculated odds for games of luck by counting up equally probable cases.
His little book, however, was not published untilby which time the elements of the theory of chances were already well known to mathematicians in Europe.
It will never be known what would have happened had Cardano published in the s. It cannot be assumed that probability theory would have taken off in the 16th century. Cardano, moreover, had no great faith in his own calculations of gambling odds, since he believed also in luck, particularly in his own.
In the Renaissance world of monstrosities, marvels, and similitudes, chance—allied to fate—was not readily naturalized, and sober calculation had its limits.
It was, for example, used by the Dutch mathematician Christiaan Huygens in his short treatise on games of chance, published in Huygens refused to define equality of chances as a fundamental presumption of a fair game but derived it instead from what he saw as a more basic notion of an equal exchange.
Most questions of probability in the 17th century were solved, as Pascal solved his, by redefining the problem in terms of a series of games in which all players have equal expectations.
The new theory of chances was not, in fact, simply about gambling but also about the legal notion of a fair contract. A fair contract implied equality of expectations, which served as the fundamental notion in these calculations.
Measures of chance or probability were derived secondarily from these expectations. Probability was tied up with questions of law and exchange in one other crucial respect.
Chance and riskin aleatory contracts, provided a justification for lending at interest, and hence a way of avoiding Christian prohibitions against usury. Lenders, the argument went, were like investors; having shared the risk, they deserved also to share in the gain.
For this reason, ideas of chance had already been incorporated in a loose, largely nonmathematical way into theories of banking and marine insurance. From aboutinitially in the Netherlands, probability began to be used to determine the proper rates at which to sell annuities.
Jan de Wit, leader of the Netherlands from tocorresponded in the s with Huygens, and eventually he published a small treatise on the subject of annuities in Annuities in early modern Europe were often issued by states to raise money, especially in times of war. This formula took no account of age at the time the annuity was purchased.
Wit lacked data on mortality rates at different ages, but he understood that the proper charge for an annuity depended on the number of years that the purchaser could be expected to live and on the presumed rate of interest. Despite his efforts and those of other mathematicians, it remained rare even in the 18th century for rulers to pay much heed to such quantitative considerations.
Life insurance, too, was connected only loosely to probability calculations and mortality records, though statistical data on death became increasingly available in the course of the 18th century. The first insurance society to price its policies on the basis of probability calculations was the Equitable, founded in London in But from medieval times to the 18th century and even into the 19th, a probable belief was most often merely one that seemed plausible, came on good authority, or was worthy of approval.
Probability, in this sense, was emphasized in England and France from the late 17th century as an answer to skepticism.Blaise Pascal (/ p æ ˈ s k æ l, p ɑː ˈ s k ɑː l /; French: [blɛz paskal]; 19 June – 19 August ) was a French mathematician, physicist, inventor, writer and Catholic theologian.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Probability and statistics: Probability and statistics, the branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data.
Learn more about the history of probability and statistics in this article. It took him until the age of thirty to finally perfect the machine, which was known at the Pascaline. It makes mathematics a lot easier to understand and is one of the most important tools used by people.
Blaise Pascal (–) Blaise Pascal was a French philosopher, mathematician, scientist, inventor, and theologian. In mathematics, he was an early pioneer in .
Pascal’s Wager, written by Blaise Pascal, in essence states that it is prudent to believe in God’s existence because it is the best bet.
Even if one assumes, that God’s existence is extremely unlikely, betting on it makes sense, for its results far outweighs the results from not betting on God’s existence. - Blaise Pascal Blaise Pascal was a French mathematician, physicist, and religious philosopher. He had many important contributions to the mathematics and physics such as: the construction of mechanical calculators, considerations on probability theory, the study of fluids, concepts of the pressure and vacuum, and the Pascal Triangle.