Question

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.

Accepted Answer

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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Concetti chiave

Geometric Distribution

Binomial Distribution

Unusual Events