A Probability Course for the Actuaries: A Preparation for by Marcel B. Finan

By Marcel B. Finan

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Circular permutations. However, if the two above permutations n are counted as the same, then the total number of circular permutations of . this type is (n−1)! 6 In how many ways can you seat 6 persons at a circular dinner table. Solution. There are (6 - 1)! = 5! = 120 ways to seat 6 persons at a circular dinner table. Combinations In a permutation the order of the set of objects or people is taken into account. However, there are many problems in which we want to know the number of ways in which k objects can be selected from n distinct objects in arbitrary order.

1 Given n objects of which n1 indistinguishable objects of type 1, n2 indistinguishable objects of type 2, · · · , nk indistinguishable objects of type k where n1 + n2 + · · · + nk = n. Then the number of distinguishable permutations of n objects is given by: n! n2 ! · · · nk ! Proof. The task of forming a permutation can be divided into k subtasks: For the first kind we have n slots and n1 objects to place such that the order is unimportant. There are C(n, n1 ) different ways. For the second kind we have n − n1 slots and n2 objects to place such that the order is unimportant.

Various probability concepts exist nowadays. A widely used probability concept is the experimental probability which uses the relative frequency of an event and is defined as follows. Let n(E) denote the number of times in the first n repetitions of the experiment that the event E occurs. Then P (E), the probability of the event E, is defined by n(E) . n→∞ n This states that if we repeat an experiment a large number of times then the fraction of times the event E occurs will be close to P (E). 1.

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