MEDAL REVIEW Waiting periods, instructive signals and positional information

Two former biologists play at dice. In the center of the table there are several banknotes from a prize they had won a few years before they dropped out of science. The rule of the game is that each player gets a banknote whenever he correctly predicts how many throws it will take after throwing a 6 to throw the next 6. One of the two players, a former theoretical biologist, remembers that the frequency of throwing a 6 is one in six, so he always foretells that the waiting period will be 6. The other player’s cause for failing in science was opposite: he believed in superstitions. As his lucky number is three, he guesses after each 6 that the next 6 will occur three throws later. Which of the two fellows will recover more from the prize money? And is there a waiting period that could be predicted that would make more money?