Testing for Favorable Numbers on a Roulette Wheel

We are forced to accept as [an] alternative that the random spinning of a roulette manufactured and daily readjusted with extraordinary care is not obedient to the laws of chance, but is chaotic in its manifestations! (Karl Pearson 1894) Given integers k ≥ k 0 ≥ 2, a test of fixed sample size and a sequential test are constructed for the purpose of testing the null hypothesis that max1≤i≤k p i ≤ 1/k 0, against the alternative that max1≤i≤k p i > 1/k 0, where p 1, …, pk are the parameters of a multinomial distribution with k cells. Results are applied to the problem of testing for favorable numbers on a roulette wheel, in which case k = 38 and k 0 = 36, thereby providing a partial solution to a problem posed by Wilson (1965).