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By William Feller

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A party of six people goes to the theatre. In how many ways may they be seated in six adjacent seats? 3. Suppose three men and three women from a singles club are to be seated in the six seats of exercise 2 in such a way that two people of the same sex never sit next to each other. In how many ways can this be done? 4. Suppose I have seven adjacent spaces and I want to fill the spaces with the letters H and T such that I write the letter H four times and the letter T three times. How many distinguishable patterns can be formed in this way?

A symmetry is now apparent in this model: any ordered pair with R as the first coordinate corresponds to an ordered pair with R in the second coordinate by interchanging the first and second coordinates. The number of outcomes in the events Al and A2 are therefore the same, and so we should not now find it surprising that they have the same probability. This way of thinking about the problem also shows that we should expect the same phenomenon at the nth selection if there are enough balls in the urn to make n selections, that is, the probability of selecting a red ball at selection n is still a / (a + b).

The Bayesian views P{A) on the right as an original or prior subjective probability, and the conditional probability on the left as the updated, or posterior, version of this subjective probability using the additional information supplied by B. The main criticisms of the Bayesian approach concern the need for assuming the prior probability distribution and the method of determining it. On the other hand, those who want to stick with repeatable events and relative frequencies can also be criticized for unnecessarily restricting the notion of probability.

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An introduction to probability theory and its applications by William Feller


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