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 is approximately 0.80 or 80%.

Step-by-step solution

Define Events and Probabilities

Let A be the event that a person is guilty of cheating (seeking illegal deductions) and B be the event that a person denies knowledge of the error when confronted. We are given the following probabilities: - P(A) = 0.05 (5% of people are guilty of cheating) - P(A') = 0.95 (95% of people are not guilty of cheating, either because they don't have any unwarranted deductions or because they are making mistakes out of unfamiliarity with regulations). - P(B|A) = 0.80 (80% of guilty people will deny knowledge of the error) We are asked to find P(A|B), the probability that a person who denies knowledge of the error is guilty of cheating.

Apply Bayes' Theorem

We can use Bayes' Theorem to find P(A|B). Bayes' Theorem is defined as follows: $$ P(A|B) = \frac{P(B|A) P(A)}{P(B)} $$ We have P(B|A) and P(A), but we need to find P(B) (the probability that a person denies knowledge of the error).

Find P(B)

We can find P(B) using the law of total probability. This law states that the total probability of an event B can be found by summing the conditional probabilities P(B|A) and P(B|A'): $$ P(B) = P(B|A)P(A) + P(B|A')P(A') $$ We have P(B|A) = 0.80 and P(A) = 0.05. We need to find P(B|A') (the probability that a person who is not guilty will deny knowledge of the error). We are not given this value directly, but we know that 2% of people make mistakes unintentionally. Since the total percentage of people who are not guilty is 95% (P(A') = 0.95), 2% of those 95% can represent the group that makes mistakes. Thus, we can assume that P(B|A') = 0.02/0.95.

Calculate Probability

Now that we have the necessary probabilities, we can use Bayes' Theorem to calculate P(A|B): $$ P(A|B) = \frac{P(B|A) P(A)}{P(B|A)P(A) + P(B|A')P(A')} $$ Now plug in the values: $$ P(A|B) = \frac{0.80*0.05}{0.80*0.05 + \frac{0.02}{0.95}*0.95} $$ Simplify the equation: $$ P(A|B) = \frac{0.80*0.05}{0.80*0.05 + 0.02} $$ Compute the result: $$ P(A|B) \approx 0.80 $$

Conclusion

The probability that a person who denies knowledge of the error when confronted with an unwarranted deduction is guilty is approximately 0.80 or 80%.

Key concepts

Probability

Probability is a branch of mathematics that deals with the likelihood of occurrences of different events. It gives us a way to quantify uncertainty. In the context of this exercise, probability helps to identify how likely it is that certain events, such as filing illegal deductions, will happen. Here are some basics to understand:

• Probability of an event: Usually represented by P(Event), it ranges from 0 to 1, where 0 means the event will not occur, and 1 means the event will certainly occur.
• Complementary Events: If there is an event A, the complement, A', represents the event A not occurring. The sum of probabilities of complementary events always equals 1.

For instance, in our exercise, we are told 5% of people are guilty of cheating. This probability is expressed as P(A) = 0.05. Conversely, the probability that someone is not guilty is given as P(A') = 0.95, since either someone is guilty or not guilty.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. It’s notated as P(B|A), representing the probability of event B occurring given event A has occurred. This concept is crucial in situations where one event influences the probability of another. In the given exercise, we use conditional probability to understand behaviors based on certain conditions. For example:

• P(B|A) = 0.80 is the probability that someone who is guilty of cheating will deny the error if confronted.
• The aim is to find P(A|B), which is the probability someone is guilty given they have denied knowledge of the error.

Conditional probabilities unlock the understanding of how likely events are to happen based directly on the presence of other conditions or events. It is an essential part of statistical inference and crucial for applications involving Bayes' Theorem.

Law of Total Probability

The law of total probability provides a way to break down a complicated probability problem into simpler parts. This law is particularly useful when the total probability of an event is influenced by several distinct scenarios. In this exercise, the law of total probability helps calculate P(B), the total probability of denying the error. This is accomplished by considering different paths (or scenarios) that lead to the event of interest:

• P(B) is calculated as: P(B|A)P(A) + P(B|A')P(A').
• In this context, it means combining the probabilities of denying the error by both guilty and non-guilty individuals.

By applying this law, one can compute P(B), which is then used in Bayes' Theorem to find conditional probabilities like P(A|B). It's a bridge to bring all contributing probabilities together to form a complete probability picture.