In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
One out of every 42 tax returns for incomes over $1 million requires an audit. An auditor is examining tax returns for over $1 million. Find the probability that (c) none of the first five returns the tax auditor examines require an audit.

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Step 1: Identify the type of probability distribution to use. Since the problem involves repeated independent trials with a fixed probability of success (a tax return requiring an audit), the binomial distribution is appropriate. The binomial distribution is defined as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

Step 2: Define the parameters of the binomial distribution. Here, n = 5 (the number of tax returns examined), k = 0 (we are looking for the probability of no audits), and p = 1/42 (the probability of a single tax return requiring an audit).

Step 3: Substitute the values into the binomial probability formula. Using MathML, the formula becomes:

P (X = 0) = (n! / (k! (n - k)!)) * p^k * (1 - p)^(n - k)

. For this problem, substitute n = 5, k = 0, and p = 1/42.

Step 4: Simplify the formula. Since k = 0, the term p^k becomes 1, and the binomial coefficient simplifies to 1. The formula reduces to P(X = 0) = (1-p)^n. Substitute p = 1/42 and n = 5 into the formula:

P (X = 0) = (1 - 1 / 42)^5

Step 5: Determine whether the event is unusual. An event is typically considered unusual if its probability is less than 0.05. Calculate the probability from Step 4 and compare it to 0.05 to determine if the event is unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. In this context, it can be used to find the probability of observing a certain number of failures before the first success occurs. The probability of success is constant across trials, making it suitable for scenarios like auditing tax returns.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it can be applied to determine the probability of none of the five tax returns requiring an audit, where each return has a success probability of 1/42. The binomial formula is useful for calculating probabilities in scenarios with a defined number of trials.

Unusual Events

An event is considered unusual if its probability is low, typically defined as less than 5%. In the context of this problem, after calculating the probability of none of the five tax returns requiring an audit, one must assess whether this probability falls below the threshold for being classified as unusual. This helps in understanding the significance of the results in practical terms.

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