91Ó°ÊÓ

Calculate the Bernoulli numbers \(B_{8}, B_{10}\), and \(B_{12} .\) The sequence of Bernoulli numbers is usually completed by setting \(B_{0}=1, B_{1}=-\frac{1}{2}\), and \(B_{k}=0\) for \(k\) odd and greater than 1 .

Show that if an event of unknown probability happens \(n\) times in succession, the odds are \(2^{n+1}-1\) to 1 for more than an even chance of its happening again.