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Can the probability of an event ever be a transcendental number?
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In general,prob.s are not rational no.s.So then is it true that probabilites are not represented as ratios of a given sample space?
If you take Kolmogorov's axiomatic approach or in general think of prob. as a measure then they are not necessarily rational.I try to understand why a probability can be extended to be anything more than that of a rational number.
It is true that P(A) can be thought of (always) as m(events that result in A)/m(total number of events), where m is a measure. Your mistake is in thinking that all probability measures come from events, and state spaces, with a finite number of elements, and that the measure is just counting the number of elements in each set.So then is it true that probabilites are not represented as ratios of a given sample space?
You are limiting your understanding of probability to a finite number of equally-likely outcomes -- drawing a card, flipping a coin, rolling dice. This is the classical definition of probability developed by Laplace. The concepts of probability and statistics have long since been extended to cover far more than gambling. For example, what is the probability it will rain tomorrow, or that the light bulb you just replaced will fail within a week? Classical probability theory cannot answer these questions, as the spaces are not enumerable. Modern probability theory can answer these kinds of questions.So then is it true that probabilites are not represented as ratios of a given sample space?
I try to understand why a probability can be extended to be anything more than that of a rational number.
That is a beauty!MeJennifer: Can the probability of an event ever be a transcendental number?
There is a famous math problem from 1777, not long after Laplace in 1771 begain submitting papers on probability; this problem does fit the bill: Buffon's Needle Problem.
Here, we draw parallel lines on the floor, d apart, and drop a needle of length l on the floor. How likely is it that the needle will land on a line? For l=d, the needle being as long as the distance between the lines, the answer is 2/pi =.6366
http://mathworld.wolfram.com/BuffonsNeedleProblem.html