Sometimes the outcomes in a probability experiment are not equally likely. For instance, in the spinner below, the outcomes blue and yellow are equally likely because they represent the same area on the spinner, but the outcome red is twice as likely because it occupies twice as much area as either blue or yellow. The sample space is S = {blue, red, yellow} even though each outcome is not equally likely.
But to simplify the problem for this case, we could rewrite the sample space as {blue, red1, red2, yellow}. By doing this, each of the outcomes listed is equally likely because we have listed the color red twice. Notice that because we were writing this as a set, we cannot simply write red in the set twice, because each element in a set must be distinct or it would represent the same element. By listing the outcomes as red1 and red2, we are indicating that there are two distinct areas that result in the outcome of red as illustrated in the diagram below.
Note: Often a sample space does not have its outcomes all equally-likely. Further, we are often not able to do the above procedure where the outcomes are made equally-likely.
He who has heard the same thing told by 12,000 eye-witnesses has only 12,000 probabilities, which are equal to one strong probability, which is far from certain.
Voltaire (1694-1778)In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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Date last modified: June 18, 2010.
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