The probability that the IRS auditor selects none of the three tax returns containing errors is calculated by multiplying the probabilities of selecting returns without errors sequentially. The first return has a probability of 30 out of 45 not containing errors, the second has 29 out of 44, and the third has 28 out of 43. This results in a final probability of (2/3)(29/44)(28/43), which can be rounded to four decimal places for precision. Additionally, the probability that a randomly selected tax return contains errors is 15 out of 45, while the probability that it does not contain errors is 30 out of 45. The discussion emphasizes the importance of understanding independent probabilities in this context.