On a television game show, a contestant is offered a sequence of prizes, which are independent and identically distributed random variables taking possible values \(\$ 1000, \$ 2000\), and \(\$ 3000\) with probabilities \(1 / 2,1 / 8\), and \(1 / 8\), respectively. The contestant may at any time choose to accept a prize, at which point the game is over. The game show host may, at some point, choose to offer no more prizes, and the contestant departs with nothing. This happens with probability \(1 / 4\). When should the contestant accept a prize?