The chance of an IRS audit for a tax return with over 25000

The chance of an IRS audit for a tax return with
over $25,000 in income is about 2% per year. We are interested in the expected
number of audits a person with that income has in a 20 year period. Assume
each year is independent.
i) How many audits are expected in a 20 year period?
ii) Find the probability that a person is not audited at all.
iii) Find the probability that a person is audited more than twice.
Suppose that 100 people with tax returns over $25,000 are randomly picked.
We are interested in the number of people audited in 1 year.
iv) How many are expected to be audited?
v) Find the probability that no one was audited.
vi) Find the probability that more than 2 were audited.

Solution

X B(20, 0.02) or we could use the mean value calculated below and approximate with the Poisson distribution X P(.4).

i)How many audits are expected in a 20 year period? Solution: µX = np = 20(0.02) = 0.4

ii). Find the probability that a person is not audited at all. Solution: (.98)20 or binompdf(20, 0.02, 0) = 0.668 or poissonpdf(.4, 0) = .670.

iii. Find the probability that a person is audited more than twice. Solution: 1 binomcdf(20, 0.02, 2) = 0.007 or 1poissoncdf(.4, 2) = .008

One way to solve this problem is by using the Binomial Distribution. Since n is large and p is small, another discrete distribution could be used to solve the following problems.