Question 1199633: VERCOR provides merger and acquisition consultants to assist corporations when owners decide to offer their business for sale. One of its news releases, “Tax Audit Frequency Is Rising,” written by David L. Perkins Jr., a VERCOR partner and which originally appeared in The Business Owner, indicated that the proportion of the largest businesses, those corporations with assets of $10 million and over, that were audited was 0.17.
a. One member of VERCOR’s board of directors is on the board of directors of four other large corporations. Calculate the expected number of these five corporations that should get audited, assuming selection is random.
b. Three of the five corporations were actually audited. Determine the probability that at least three of the five corporations would be audited if 17% of large corporations are audited. (Assume random selection.)
c. The board member is concerned that the corporations have been singled out to be audited by the Internal Revenue Service (IRS). Respond to these thoughts using probability and statistical logic
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a) Expected Number of Audited Corporations**
* **Probability of Audit:** 0.17
* **Number of Corporations:** 5
* **Expected Number of Audited Corporations:**
* Expected Number = Probability of Audit * Number of Corporations
* Expected Number = 0.17 * 5 = 0.85
**Therefore, the expected number of audited corporations is 0.85.**
**b) Probability of at Least Three Audits**
* This is a binomial probability problem.
* **n = 5** (number of trials - corporations)
* **p = 0.17** (probability of success - audit)
* We need to find: P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)
* **Binomial Probability Formula:**
* P(X = k) = (nCk) * p^k * (1-p)^(n-k)
* where nCk = n! / (k! * (n-k)!)
* **Calculate Probabilities:**
* P(X = 3) = (5C3) * (0.17)^3 * (0.83)^2 ≈ 0.0284
* P(X = 4) = (5C4) * (0.17)^4 * (0.83)^1 ≈ 0.0024
* P(X = 5) = (5C5) * (0.17)^5 * (0.83)^0 ≈ 0.000014
* **Calculate P(X ≥ 3):**
* P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)
* P(X ≥ 3) ≈ 0.0284 + 0.0024 + 0.000014
* P(X ≥ 3) ≈ 0.0308
**Therefore, the probability of at least three of the five corporations being audited is approximately 0.0308.**
**c) Addressing the Board Member's Concerns**
The board member's concern that the corporations are being "singled out" for audits can be addressed using probability and statistical logic:
* **Calculate the expected number of audits:** As calculated in part (a), the expected number of audited corporations is 0.85.
* **Compare actual and expected:** Three corporations were audited, which is higher than the expected value of 0.85.
* **Consider the probability of observing three or more audits:** As calculated in part (b), the probability of three or more audits occurring by chance is 0.0308. This is a relatively low probability.
**While three audits out of five might seem unusual, it's important to remember that:**
* **Random events can be surprising:** Even with a low probability, unusual events can still occur randomly.
* **Small sample size:** With only five corporations, it's difficult to make strong conclusions about whether the observed number of audits is truly unusual.
* **Other factors may influence audit selection:** Factors like the complexity of the companies' financial statements, the industry they operate in, and recent tax law changes could also influence the IRS's audit selection process.
**To further investigate the board member's concerns:**
* **Gather more data:** Analyze audit rates for similar companies over a longer period.
* **Consult with tax experts:** Seek advice from tax professionals who can assess the specific circumstances of these corporations and provide expert opinions.
**Disclaimer:** This analysis provides a basic framework for addressing the board member's concerns. It's crucial to consult with qualified tax professionals for personalized and comprehensive advice.
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