The IRS will do a random sampling of tax returns filed in 2010. They are concerned about major fraud, and have a new electronic evaluation procedure designed to detect it. Examination of tax returns for such "Fraudulent" ones is regarded as a series of independent trials (or, repetitions of the same experiment), shere they believe that 1/3 of all returns filed are "Fraudulent".

Base all your calculations for the current problem/experiment/investigation on this model.

a) Let K denote the number of fraudulent tax returns among the first 6 returns examined. Rename the rv K, name its distribution, and then evaluate algebraically the probability that K=4. Evaluate algebraically and numerically the mean and variance of K.

b) Let V denote the number of tax returns that will be examined until the 16th fraudulent return is discovered. Rename the rv V, name its distribution, and then evaluate algebraically the probability that it takes exactly 50 trials to find the 16th fraudulent return.

Evaluate algebraically and numerically the mean and variance of K.