Question

suppose 1.5% of people who file federal tax returns with an adjusted gross income (agi) between $75,000 and $100,000 are audited. of 5 people in that tax bracket for whom abc tax guys prepared their taxes, 2 were audited.

a. if 5 people with an agi between $75,000 and $100,000 are selected at random from all the people who filed federal tax returns, what is the probability that at least 2 are audited?

b. would you recommend that a friend with an agi between $75,000 and $100,000 use abc tax guys to prepare her returns? explain.

a. the probability is about
(type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Here, \(n = 5\), \(p=0.015\) (since \(1.5\%=0.015\)), and we want \(P(X\geq2)=1 - P(X = 0)-P(X = 1)\)

Step2: Calculate \(P(X = 0)\)

$$
LATEXBLOCK0
$$

Using a calculator, \((0.985)^{5}\approx0.927\)

Step3: Calculate \(P(X = 1)\)

$$
LATEXBLOCK1
$$

\((0.985)^{4}\approx0.941\), so \(P(X = 1)=5\times0.015\times0.941 = 0.0706\)

Step4: Calculate \(P(X\geq2)\)

$$
LATEXBLOCK2
$$

Answer:

a. The probability is about \(0.002\)

b. The probability that at least 2 out of 5 people in this income - tax bracket are audited is extremely low (\(P(X\geq2)\approx0.002\)). But in the case of ABC Tax Guys, 2 out of 5 were audited. This suggests that the probability of audit for clients of ABC Tax Guys is much higher than the general probability (\(1.5\%\)) for people in the \(\$75000-\$100000\) AGI bracket. So, I would not recommend that a friend with an AGI between \(\$75000\) and \(\$100000\) use ABC Tax Guys to prepare her returns.