# Mathematics: Probability

Part 1| 2|

Part 2| 5|

Part 3| 6|

Part 4| 8|

Part 5| 10|

Further Exploration| 13|

Conclusion| 14|

Reflection| 15|

Part 1

a) Introduction

The word Probability derives from probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which, in contrast, is used as a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference. A short history of Probability Theory............

The branch of mathematics known as probability theory was inspired by gambling problems. The earliest work was performed by Girolamo Cardano (1501-1576) an Italian mathematician, physician, and gambler. In his manual Liber de Ludo Aleae, Cardano discusses many of the basic concepts of probability complete with a systematic analysis of gambling problems. Unfortunately, Cardano's work had little effect on the development

of probability because his manual, which did not appeared in print until 1663, received little attention. In 1654, another gambler named Chevalier de Méré created a dice proposition which he believed would make money. He would bet even money that he could roll at least one 12 in 24 rolls of two dice. However, when the Chevalier began losing money, he asked his mathematician friend Blaise Pascal (1623-1662) to analyze the proposition. Pascal determined that this proposition will lose about 51% of the time. Inspired by this proposition, Pascal began studying more of these types of problems. He discussed them with another famous mathematician, Pierre de Fermat (1601-1665) and together they laid the foundation of probability theory. Examples / Application & Importance of Probability Theory to real life situations................ Probability theory is concerned with determining the relationship between the number of times a certain event occurs and the number of times any event occurs. For example, the number of times a head will appear when a coin is flipped 100 times. Determining probabilities can be done in two ways; theoretically and empirically. The example of a coin toss helps illustrate the difference between the two approaches. Using a theoretical approach, we reason that in every flip there are two possibilities, a head or a tail. By assuming each event is equally likely, the probability that the coin will end up heads is 1/2 or 0.5. The empirical approach does not use assumption of equal likeliness. Instead, an actual coin flipping experiment is performed and the number of heads is counted. The probability is then equal to the number of heads divided by the total number of flips. Two major applications and importance of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis", often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. Accordingly, it may be of some importance to most citizens to understand how odds...

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