In discrete distributions, it has often been recommended that an auxiliary random experiment should be carried out so as to make the size of the test equal to the significance level. In certain experimental situations, this procedure would be time-consuming and even embarrassing to the statistician. An alternative criterion of significance is suggested, the mid or median probability. To avoid computationsP(χ), the probability of the square root of a chi-square variable, can be used in the binomial, Poisson and hypergeometric distributions as an approximation to the mid probability. In statistical control of counting experiments or where experiments are being repeatedP(χ) rather thanP(χc), the probability of χ corrected for continuity, gives acceptable approximations to size of significance levels. Some computations are given for the multinomial distribution which show that hereP(χ2) gives acceptable approximations.