Question

According to the Internal Revenue Service (IRS), the chances of your tax return being audited are about 6 in 1,000 if your income is less than $50,000; 10 in 1,000 if your income is between $50,000 and $99,999; and 49 in 1,000 if your income is $100,000 or more (Statistical Abstract of the United States: 1995).

If two taxpayers with incomes under $50,000 are randomly selected and two with incomes more than $100,000 are randomly selected, what is the probability that none of these taxpayers will be audited by the IRS?

Answer 1

Given, according to the Internal Revenue Services ,the chances of account being audited are :

Income		Chances of Audit		Probability of audit
<50000		6 in 1000		6/1000 = 0.006
>50000 and <99999		10 in 1000		10/1000 = 0.01
>= 100000		49 in 1000		49/1000 = 0.049

Two tax payers with income under 50,000 are randomly seloected and two with income greater than 100,000.

Note that the chances of any tax payer being audited is completely independent of the chances of any other tax payers being audited.

Hence, the probability that none of them are audited =

(Probability of tax payer with income <50000 not being audited ) ²  * (Probability of tax payer with income 100000 not being audited )²

From the table above, the probability of any tax payer with income <50000 not being audited

= 1 - 0.006 = 0.994

And, the probability of any tax payer with income > 100000 not being audited

= 1 - 0.049 = 0.951

Hence, the required probability is  

= 0.994²  * 0.951²

= 0.09880 * 0.904401

= 0.08935 (approx)

So, the probability that none of the taxpayers will be audited by IRS = 0.08935