Two statistical coverage problems in estimating the variance of a population

Suppose a researcher draws a sample X1, X2, . . . , Xm from some population, and computes the corresponding variance in it. This in order to estimate the variance of the population from which the sample was drawn. Assume that the population in question has a Gaußian probability distribution. A second researcher draws, independently, a sample Y1,Y2, . . . ,Yn from the same population. He computes not only the corresponding variance in the sample, but also surrounds it by margins such as to get a 95% confidence interval for the population variance. Then what is the probability that this 95% confidence interval, generated by Y1,Y2, . . . ,Yn , will cover the sample variance of X1, X2, . . . , Xm? Below this probability will be denoted by Pm n . As a second coverage problem, what is the probability that the 95% confidence interval generated by the X1, X2, . . . , Xm and that by the Y1,Y2, . . . ,Yn are disjoint? Below this probability will be denoted by Qn . The aim