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 (b) the first return requiring an audit is the first or second return the tax auditor examines, 

Verified step by step guidance

1. Step 1: Identify the type of probability distribution to use. Since the problem involves finding the probability of the first success (a tax return requiring an audit) occurring within a specific number of trials, this is a geometric distribution problem.
2. Step 2: Define the parameters of the geometric distribution. The probability of success (p) is given as 1/42, since one out of every 42 tax returns requires an audit. The probability of failure (q) is therefore 1 - p = 41/42.
3. Step 3: Write the formula for the geometric probability distribution. The probability that the first success occurs on the k-th trial is given by: P(X = k) = q^(k-1) * p, where X is the random variable representing the trial on which the first success occurs.
4. Step 4: Calculate the probability for the first return requiring an audit (k = 1) and the second return requiring an audit (k = 2). Use the formula P(X = k) = q^(k-1) * p for each case. For k = 1, P(X = 1) = (41/42)^0 * (1/42). For k = 2, P(X = 2) = (41/42)^1 * (1/42).
5. Step 5: Add the probabilities for k = 1 and k = 2 to find the total probability that the first return requiring an audit is the first or second return. That is, P(X = 1 or X = 2) = P(X = 1) + P(X = 2).

Key Concepts

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, a 'success' is defined as finding a tax return that requires an audit. The probability of success on each trial is constant, and the distribution is characterized by the formula P(X = k) = (1-p)^(k-1) * p, where p is the probability of success.

Probability Calculation

Calculating probabilities involves determining the likelihood of a specific event occurring. In this case, we need to find the probability that the first return requiring an audit occurs on the first or second return examined. This requires summing the probabilities of the first success occurring on the first trial and the second trial, which can be computed using the geometric distribution formula.

Unusual Events

An event is considered unusual if its probability is significantly low, typically defined as less than 5%. In the context of this problem, after calculating the probability of the first return requiring an audit being the first or second examined, we would assess whether this probability falls below the 5% threshold to determine if the event is unusual.