prob.part32.65_66 - 2.2. CONTINUOUS DENSITY FUNCTIONS 57...

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2.2. CONTINUOUS DENSITY FUNCTIONS 57 game there may well be!), it is natural to assume that the coordinates are chosen at random. (When doing this with a computer, each coordinate is chosen uniformly from the interval [ - 1 , 1]. If the resulting point does not lie inside the unit circle, the point is not counted.) Then the arguments used in the preceding example show that the probability of any elementary event, consisting of a single outcome, must be zero, and suggest that the probability of the event that the dart lands in any subset E of the target should be determined by what fraction of the target area lies in E . Thus, P ( E ) = area of E area of target = area of E π . This can be written in the form P ( E ) = ± E f ( x ) dx , where f ( x ) is the constant function with value 1 . In particular, if E = { ( x, y ) : x 2 + y 2 a 2 } is the event that the dart lands within distance a < 1 of the center of the target, then P ( E ) = πa 2 π = a 2 . For example, the probability that the dart lies within a distance 1/2 of the center is 1/4. ± Example 2.9
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prob.part32.65_66 - 2.2. CONTINUOUS DENSITY FUNCTIONS 57...

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