Question

The Internal Revenue Service audits a random sample of 60% of tax returns where the gross income exceeds $100,000. If a tax lawyer assists in the completion of 5 such returns, what is the probability that between 2 and 4 will be audited.

Answer 1

Percentage of tax returns where  the gross income exceeds $100,000 that are being audited = 60%

Proabability of tax return being audited : p = 60/100 = 0.6

q = 1-p = 1-0.6 = 0.4

Number of completed returns : n= 5

'X' - Number of completed returns that will be audited

Probability that between 2 and 4 completed returns will be audited = P(2X4) = P(X=2)+P(X=3)+P(X=4)

The above probability is computed using Binomail distribution.

If 'X' is the random variable representing the number of successes, the probability of getting ‘r’ successes and ‘n-r’ failures, in 'n' trails, ‘p’ probability of success ‘q’=(1-p) is given by the probability function

Applying binomial distribution to the given problem,

Probability that 'r' tax returns will be audited from 5 completed returns

P(2X4) = P(X=2)+P(X=3)+P(X=4) = 0.2304 + 0.3456 + 0.2592 = 0.8352

Probability that between 2 and 4 completed returns will be audited = 0.8352