Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A tax specialist has estimated that the probability that a tax return selected at random will be audited is .02. Furthermore, he estimates that the probability that an audited return will result in additional assessments being levied on the taxpayer is .60. What is the probability that a tax return selected at random will result in additional assessments being levied on the taxpayer?

Step-by-Step Solution

Answer

The probability that a tax return selected at random will result in additional assessments being levied on the taxpayer is \(0.012\) or \(1.2\%\).

Step 1: Identify the given probabilities

The problem gives us the following probabilities: 1. The probability of a tax return being audited (P(Audit)) is 0.02. 2. The probability of an audited return resulting in additional assessments (P(Additional Assessments | Audit)) is 0.60.

Step 2: Apply the conditional probability formula

We can calculate the probability of a tax return resulting in additional assessments (P(Additional Assessments)) using the conditional probability formula: P(Additional Assessments) = P(Additional Assessments | Audit) * P(Audit)

Step 3: Substitute the given probabilities in the formula

Now, substitute the given probabilities into the formula: P(Additional Assessments) = 0.60 * 0.02

Step 4: Calculate the probability

Perform the multiplication: P(Additional Assessments) = 0.012

Step 5: Interpret the result

The probability that a tax return selected at random will result in additional assessments being levied on the taxpayer is 0.012 or 1.2%.

Key Concepts

Conditional Probability

Conditional probability is an important concept in probability theory that tells us the likelihood of one event occurring given that another event has already occurred. It can help us work through complex scenarios by breaking down probabilities for parts of the whole. Consider it like asking the question, "On the condition that I'm holding an umbrella, what's the chance it's raining?"

The formula for calculating conditional probability is given by:

• \( P(B|A) = \frac{P(A \cap B)}{P(A)} \)

This reads as the probability of event B occurring given that event A has occurred is equal to the probability of both events occurring together divided by the probability of event A alone.

In the case of tax audits, we used conditional probability to find out how likely it is for additional assessments to occur given a return is audited. Recalling the formula, "|" stands for "given that," which is key when navigating these sorts of problems efficiently.

Multiplication Rule

Multiplication rule in probability is a method for finding the probability of two independent events happening together. It's often used when dealing with sequential events or when events have conditional probabilities.

We are interested in how events are related, for instance, in tax audits where you may want to know both the chances a return is audited and that it leads to additional assessments. The multiplication rule helps here:

• For independent events: \( P(A \text{ and } B) = P(A) \times P(B) \)
• For dependent events: \( P(A \text{ and } B) = P(A) \times P(B|A) \)

In our original exercise, since we utilized conditional probability, event dependencies are accounted for by directly joining probabilities, forming one composite probability. The multiplication rule simplifies the process by linking them to perform just one calculation, as in stepping through: \( 0.60 \times 0.02 = 0.012 \).

This method not only breaks down complexities but also offers a clear way to interpret interrelated probabilities.

Tax Audit Probabilities

Tax audit probabilities focus specifically on understanding and calculating the chances involved in tax audits. These probabilities help individuals, businesses, and tax specialists to evaluate the risks associated with tax filings being audited and the subsequent consequences.

Considering the original exercise, we dealt with these probabilities:

• The general chance of a tax return being selected for audit: \( 0.02 \), or 2%.
• The likelihood that an audited return will lead to additional assessments: \( 0.60 \), or 60%.

The combination of these two gives the overarching probability of additional assessments following an audit, derived as 0.012 or 1.2%.

This analysis allows tax specialists to know that even though the general audit rate is low, the substantial probability of further assessments makes precise filing all the more essential. Understanding audit probabilities creates better preparedness for both filing taxes and facing possible outcomes, ultimately aiding in more strategic tax management.