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Extra resources for A Probability Course for the Actuaries: A Preparation for Exam P 1
Fred wants to choose 5 things from 21 things, where order doesn’t matter and repetition is allowed. 3 gives the number of vectors (n1 , n2 , · · · , nk ), where ni is a nonnegative integer, such that n1 + n2 + · · · + nk = n where ni is the number of objects in box i, 1 ≤ i ≤ k. 8 How many solutions to x1 + x2 + x3 = 11 are there with xi nonnegative integers? Solution. This is like throwing 11 objects in 3 boxes. The objects inside each box are indistinguishable. Let xi be the number of objects in box i where i = 1, 2, 3.
This 4 1 is what one gets when replacing n = 4 and k = 2 in the following theorem. e. indistinguishable) objects can be distributed into k boxes is n+k−1 k−1 = n+k−1 n . Proof. Imagine the n identical objects as n stars. Draw k−1 vertical bars somewhere among these n stars. This can represent a unique assignment of the balls to the boxes. Hence, there is a correspondence between a star/bar picture and assignments of balls to boxes. So how many ways can you arrange n identical dots and k − 1 vertical bar?
3 ‡ An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages 5 respectively. A, B, and C are 14 , 13 , and , 12 Determine the probability that a randomly chosen employee will choose no supplementary coverage. 4 An experiment consists of throwing two four-faced dice.