What is the probability that two randomly selected returns with income of​ $100,000 or more will be​ audited?

Question

For the fiscal year​ 2007, a tax authority audited

1.52​%

of individual tax returns with income of​ $100,000 or more. Suppose this percentage stays the same for the current tax year.

Answer 1

If the probability that one randomly selected return with income of​ $100,000 or more will be​ audited is 1.52%, then we can use the binomial distribution to find the probability that two randomly selected returns with income of​ $100,000 or more will be​ audited. The binomial distribution is given by the formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

where:

1. P(X = k) is the probability of getting k successes in n independent trials
2. n is the number of trials
3. k is the number of successes
4. p is the probability of success in each trial
5. (n choose k) is the binomial coefficient, which is the number of ways to choose k items from a set of n items, and is given by the formula (n choose k) = n! / (k! * (n - k)!)

We want to find the probability that two randomly selected returns with income of​ $100,000 or more will be​ audited, given that the P(x) = 1.52%. Therefore, we have:

n = 2 (we are selecting two returns)

k = 2 (we want both returns to be audited)

p = 0.0152 (the probability that one return is audited)

Using the binomial distribution formula, we have:

P(X = 2) = (2 choose 2) * 0.0152^2 * (1 - 0.0152)^(2 - 2) = 0.000231

Therefore, the probability that two randomly selected returns with income of​ $100,000 or more will be​ audited, given that the probability is 1.52%, is approximately 0.0231%.