Problem 69 - An Introduction to Mathematical Statistics and Its Applications

Question

Because of her past convictions for mail fraud and forgery, Jody has a \(30 \%\) chance each year of having her tax returns audited. What is the probability that she will escape detection for at least three years? Assume that she exaggerates, distorts, misrepresents, lies, and cheats every year.

Step-by-Step Solution

Answer

The probability that Jody will escape detection for at least three years is approximately 34.3%

Step 1: Calculating Individual Probabilities

First, determine the probability of Jody not getting audited in a year. This is equal to \(1 - 0.3\) due to the complementary rule of probability since the problem states that Jody has a 30% chance of getting audited. Therefore, the probability of not getting audited is \(0.7\) or 70%.

Step 2: Applying Rule of Multiplication for Independent Events

Next, multiply the yearly probabilities together to get the overall probability for three consecutive years, because the events are independent. This is done as follows: \(0.7 * 0.7 * 0.7 = 0.343\)

Step 3: Interpreting the Result

The final step is to interpret the result. So, the probability of Jody escaping detection for at least three years is \(0.343\) or 34.3%, assuming that she is dishonest every year and that each year is an independent event.

Key Concepts

Independent Events in Probability

Understanding how independent events work in probability is crucial for solving problems about the chance of certain outcomes happening over time, such as escaping tax audits. Independent events are those where the outcome of one event does not affect the outcome of another.

For example, if a person tosses a coin and then rolls a dice, the result of the coin toss has no influence on what number the dice lands on. These are independent events. In the case of Jody, the probability of her getting audited each year is treated as an independent event. This means the audit outcome of one year doesn't change the likelihood of getting audited in the subsequent years.

Even if Jody has been audited before, it won't affect her chances of being audited again the following year. This assumption is key in calculating probabilities over multiple years.

Complementary Rule of Probability

The complementary rule of probability is a fundamental concept used to find the probability of the opposite of a particular event. The sum of the probabilities of an event and its complement is always 1 (100%).

For instance, if there's a 30% chance of an event occurring, like Jody getting audited, there must be a 70% chance of it not occurring, because these two outcomes cover all possible scenarios.

It's calculated simply as: \[ P(\text{not audited}) = 1 - P(\text{audited}) \] where \( P \) stands for probability. Applying this concept helps us to easily find the probability of an event not happening, especially when it's more straightforward than calculating the odds of it occurring.

Multiplication Rule for Independent Events

When we're dealing with independent events and we want to find the probability of several events all occurring, we use the multiplication rule for independent events. This rule states that to find the joint probability of two or more independent events, we multiply their individual probabilities together.

To visualize this with a simple example, if there's a 50% chance it will rain on any given day, and we want to know the chance it will rain for two days in a row, we multiply the probabilities: \[ 0.5 \times 0.5 = 0.25 \] So there's a 25% chance it will rain on both days.

In Jody's scenario, to figure out the probability of her not being audited for three consecutive years, we calculate it as: \[ 0.7 \times 0.7 \times 0.7 = 0.343 \] Thus, there's a 34.3% chance she will not be audited over the span of three years given each year is independent and has a 70% chance of not being audited.