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By Marcel B. Finan

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B) The event of obtaining more than one head is the set {T HH, HT H, HHT, HHH} Probability is the measure of occurrence of an event. 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).

B) Seven of Miss Murphy’s students are girls and 5 are boys. In how many different ways can she seat the 7 girls together on the left, and then the 5 boys together on the right? 6 Using the digits 1, 3, 5, 7, and 9, with no repetitions of the digits, how many (a) one-digit numbers can be made? (b) two-digit numbers can be made? (c) three-digit numbers can be made? (d) four-digit numbers can be made? 7 There are five members of the Math Club. In how many ways can the positions of officers, a president and a treasurer, be chosen?

6 How many ways can we place 7 identical balls into 8 separate (but distinguishable) boxes? 5 PERMUTATIONS AND COMBINATIONS WITH INDISTINGUISHABLE OBJECTS51 Solution. 7 An ice cream store sells 21 flavors. Fred goes to the store and buys 5 quarts of ice cream. How many choices does he have? Solution. 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.

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A Probability Course for the Actuaries: A Preparation for Exam P 1 by Marcel B. Finan

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