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By Fearn T., Brown P.J., Besbeas P.

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The above example is not the only way of estimating the value of π by a chance experiment. 1 1 G. L. Buffon, in “Essai d’Arithm´ etique Morale,” Oeuvres Compl` etes de Buffon avec Supplements, tome iv, ed. Dum´enil (Paris, 1836). 44 CHAPTER 2. 3: Computing the area by simulation. 3 Suppose that we take a card table and draw across the top surface a set of parallel lines a unit distance apart. We then drop a common needle of unit length at random on this surface and observe whether or not the needle lies across one of the lines.

However, the decision as to which distribution function to select to describe an experiment is not a part of the basic mathematical theory of probability. The latter begins only when the sample space and the distribution function have already been defined. Determination of Probabilities It is important to consider ways in which probability distributions are determined in practice. One way is by symmetry. For the case of the toss of a coin, we do not see any physical difference between the two sides of a coin that should affect the chance of one side or the other turning up.

8, we see that P (A) = P (B) = 1 . 2 Moreover, A ∩ B = {ω3 , ω4 }, and so P (A ∩ B) = 1/4. 4, we obtain P (A ∪ B) = P (A) + P (B) − P (A ∩ B) 3 1 1 1 + − = . = 2 2 4 4 Since A ∪ B is the 6-element set, A ∪ B = {HHH,HHT,HTH,HTT,TTH,TTT} , we see that we obtain the same result by direct enumeration. ✷ In our coin tossing examples and in the die rolling example, we have assigned an equal probability to each possible outcome of the experiment. Corresponding to this method of assigning probabilities, we have the following definitions.

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