Hot-hand effects in sports and a recursive method of computing probabilities for streaks

We give a recursive method of computing probabilities associated with the waiting time to the first occurrence of a run of arbitrary length in Markovian trials of a general order. Using data from the Oakland Athletics' 2002 season as an example, we show that the assumed model order can make a large difference in computed probabilities. The algorithm is then applied to the computation of probabilities associated with strikes and nonstrikes in bowling. After showing that there is significant deviation from a model of Bernoulli trials for data from the 2003–2004 Professional Bowlers Association tour, we suggest a criterion based on the longest success run for choosing the order of Markovian dependence that gives the best fit to the streakiness characteristics of an individual bowler's data.