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

By Marcel B. Finan

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1 Consider the random experiment of rolling a die. (a) Find the sample space of this experiment. (b) Find the event of rolling the die an even number. 2 An experiment consists of the following two stages: (1) first a fair die is rolled (2) if the number appearing is even, then a fair coin is tossed; if the number appearing is odd, then the die is tossed again. An outcome of this experiment is a pair of the form (outcome from stage 1, outcome from stage 2). Let S be the collection of all outcomes.

Then there are 5 ways to fill the first spot, 4 ways to fill the third, 3 to fill the fourth, and so on. There are 5! 3 Five different books are on a shelf. In how many different ways could you arrange them? Solution. The five books can be arranged in 5 · 4 · 3 · 2 · 1 = 5! = 120 ways Counting Permutations We next consider the permutations of a set of objects taken from a larger set. Suppose we have n items. How many ordered arrangements of k items can we form from these n items? The number of permutations is denoted by P (n, k).

6 What is the probability of rolling a 3 or a 4 with a fair die? Solution. 7 In a room containing n people, calculate the chance that at least two of them have the same birthday. Solution. We have P(Two or more have birthday match) = 1 - P(no birthday match) Since each person was born on one of the 365 days in the year, there are (365)n possible outcomes (assuming no one was born in Feb 29). 5 It is important to keep in mind that the above definition of probability applies only to a sample space that has equally likely outcomes.

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