By Marcel B. Finan
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Additional info for A Probability Course for the Actuaries: A Preparation for Exam P 1
Two events A and B are said to be mutually exclusive if they have no outcomes in common. In this case A ∩ B = ∅ and P (A ∩ B) = P (∅) = 0. 2 Consider the sample space of rolling a die. Let A be the event of rolling an even number, B the event of rolling an odd number, and C the event of rolling a 2. Find (a) A ∪ B, A ∪ C, and B ∪ C. (b) A ∩ B, A ∩ C, and B ∩ C. (c) Which events are mutually exclusive? Solution.
The decision consists of two steps. The first is to select the letters and this can be done in P (26, 3) ways. The second step is to select the digits and this can be done in P (10, 4) ways. 5 How many five-digit zip codes can be made where all digits are different? The possible digits are the numbers 0 through 9. 4 PERMUTATIONS AND COMBINATIONS Solution. P (10, 5) = 10! (10−5)! 39 = 30, 240 zip codes Circular permutations are ordered arrangements of objects in a circle. Suppose that circular permutations such as are considered as different.
A) P (6, 6) = 6! = 720 different ways (b) P (6, 2) = 30 ways (c) C(6, 3) = 20 different ways As an application of combination we have the following theorem which provides an expansion of (x + y)n , where n is a non-negative integer. 4 (Binomial Theorem) Let x and y be variables, and let n be a non-negative integer. Then n C(n, k)xn−k y k n (x + y) = k=0 where C(n, k) will be called the binomial coefficient. Proof. The proof is by induction on n. Basis of induction: For n = 0 we have 0 C(0, k)x0−k y k = 1.
A Probability Course for the Actuaries: A Preparation for Exam P 1 by Marcel B. Finan