Question

Suppose \(5 \%\) of all people filing the long income tax form seek deductions that they know are illegal, and an additional \(2 \%\) incorrectly list deductions because they are unfamiliar with income tax regulations. Of the \(5 \%\) who are guilty of cheating, \(80 \%\) will deny knowledge of the error if confronted by an investigator. If the filer of the long form is confronted with an unwarranted deduction and he or she denies the knowledge of the error, what is the probability that he or she is guilty?

Short answer

Answer: The probability that a person is guilty of cheating on their income tax given that they deny the error when confronted is approximately 0.068 or 6.8%.

Step-by-step solution

Calculate the given probabilities

We are given the following probabilities: - 5% of people filing the long form are guilty of seeking illegal deductions (cheating): P(Guilty) = 0.05 - 2% of people filing the long form incorrectly list deductions (mistake): P(Mistake) = 0.02 - 80% of guilty people (cheating) will deny knowledge of the error when confronted: P(Deny | Guilty) = 0.80

Calculate P(Innocent)

Since P(Guilty) = 0.05 and P(Mistake) = 0.02, the probability of being innocent is: P(Innocent) = 1 - P(Guilty) - P(Mistake) = 1 - 0.05 - 0.02 = 0.93

Calculate P(Deny | Innocent)

Since 2% of innocent people filing the long form incorrectly list deductions (mistake), and they would most likely deny the error when confronted: P(Deny | Innocent) = 0.02

Calculate P(Deny)

Using the total probability theorem, calculate P(Deny): P(Deny) = P(Deny | Guilty) * P(Guilty) + P(Deny | Innocent) * P(Innocent) P(Deny) = (0.80 * 0.05) + (0.02 * 0.93) = 0.04 + 0.0186 = 0.0586

Calculate P(Guilty | Deny) using Bayes' theorem

Now, calculate P(Guilty | Deny) by substituting the values into Bayes' theorem: P(Guilty | Deny) = (P(Deny | Guilty) * P(Guilty)) / P(Deny) P(Guilty | Deny) = (0.80 * 0.05) / 0.0586 ≈ 0.068 Therefore, the probability that a person is guilty of cheating on their income tax given that they deny the error when confronted is approximately 0.068 or 6.8%.

Key concepts

Understanding Probability

Probability is a fundamental concept in mathematics that measures the likelihood or chance of a particular event occurring. It helps us quantify uncertainty and predict outcomes when there are various possible results. Probability values range from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty.
For instance, in the context of the income tax deductions exercise, we calculate probabilities for scenarios like a filer cheating on deductions or denying knowledge of an error when confronted.
Key points to remember about probability:

• Probability is often expressed as a fraction, decimal, or percentage.
• Real-world applications include predicting market trends, assessing risk, or even gambling.
• For decision-making, probability offers logical and quantitative reasoning.
• It can be calculated using various rules such as the addition rule or multiplication rule for independent events.

In simpler terms, understanding probability means understanding how likely it is for something to happen.

Delving into Conditional Probability

Conditional probability allows us to determine the chances of an event occurring given that another event has already occurred. This idea is central to Bayes' theorem—a powerful statistical formula used to update predictions as more evidence becomes available.
In the exercise, when a person denies knowledge of an incorrect income tax deduction, conditional probability helps us calculate the probability they are, in fact, guilty.
Key aspects of conditional probability include:

• Expressed as \( P(A|B) \), meaning the probability of event A given that B has occurred.
• Bayes' theorem incorporates conditional probability, often appearing as:\[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]
• This formula updates beliefs based on new data or evidence.
• Used across a range of fields including medicine, finance, and AI for decision-making.

Overall, conditional probability is crucial in understanding relationships between dependent events and making informed predictions based on available data.

Income Tax Deductions Simplified

Income tax deductions are incentives provided in the tax code that reduces tax liability by lowering taxable income. They can sometimes be complex, leading to honest mistakes or intentional cheating, as highlighted in the exercise.
Understanding the basics of tax deductions can lead to better financial planning and compliance with tax laws.
Important aspects of income tax deductions:

• Contribute to lowering taxable income, ultimately reducing the amount owed in taxes.
• Common deductions include mortgage interest, student loan interest, and medical expenses.
• It is crucial to understand legal vs. illegal deductions to prevent legal issues.
• Experts or tax software can be helpful in accurately filing taxes and applying eligible deductions.

Knowing the difference between legal and illegal deductions and how they impact tax returns is essential for responsible personal and business financial management.