Final answer:
The taxpayer must report $7,000 as taxable income from gambling after deducting losses from winnings. For the IRS audit question, X is a random variable representing the number of IRS audits in 20 years, where X may range from 0 to 20 and follows a binomial distribution.
Step-by-step explanation:
The statement that a taxpayer who lost $5,000 his first night in Reno and won $12,000 the second night only needs to report $7,000 as income on his tax return is true. In the United States, gambling winnings are considered taxable income, but gambling losses are allowed as a deduction only to the extent of gambling winnings. Therefore, if in the course of the tax year the taxpayer had no other gambling transactions, they could deduct the $5,000 loss from the $12,000 win, resulting in $7,000 of taxable gambling income to be reported.
Now, regarding the IRS audit related question:
a. In words, define the random variable X.
The random variable X can be defined as the number of IRS audits a person with an income of more than $25,000 experiences in a 20-year period, given that the chance of an audit in any year is 2%.
b. List the values that X may take on.
X may take on any integer value between 0 and 20, inclusive. This reflects the possibility of having no audits or being audited every year, or any number in between.
c. Give the distribution of X. X~
The distribution of the random variable X follows a binomial distribution with parameters n = 20 (number of trials or years) and p = 0.02 (probability of audit in any given year).