By William Feller

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5. S. 10 What purchases and sales would you make to ensure a sure gain for yourself? Show that a sure gain results from your choices. How much can you be sure to gain if you buy or sell no more than four tickets? 2 9 Consequences of the axioms of probability: disjoint events We now explore some consequences of coherence. , the price of certain tickets) and do not want to be a sure loser, then you are committed to the price of certain other tickets. To start, we define the complement of an event A, which we write A¯ and pronounce “not A,” to be the event that A does not happen.

N + (n + 1) = n(n + 1)/2 + (n + 1) = (n + 1)(n/2 + 1) = (n + 1)(n + 2)/2, which is S(n + 1). Therefore we have proved the second step of the induction, and have shown that the sum of the first n integers is n(n + 1)/2 for all integers n bigger than or equal to 1. Mathematical induction requires that you already think you know the solution. It is not so useful for finding the right formula in the first place. However often some experimentation and a good guess can help you find a formula which you can then try to prove by induction.

N2 ) is n(n + 1)(2n + 1)/6. 2. , 1 + 8 + 27 + . . + n3 ) is [n(n + 1)/2]2 . You can find an excellent further explanation of mathematical induction in Courant and Robbins (1958). I anticipate that most readers of this book will be familiar with at least one proof of the fact that T = n(n + 1)/2. There are two reasons for discussing it here. The first is to give a simple example of induction. The second is to show that the same mathematical fact may be approached from different directions.