Since you have posted a question with multiple sub-parts, we will solve first three sub- parts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved

(a) Determine the distribution of X.

The distribution of X is determined below as follows:

Let X denotes the number of audits a person with the income more than 25,000 which follows binomial distribution with the probability of success 0.02 and the number of years selected is 20.

That is, \(\displaystyle{n}={20},{p}={0.02},{q}={0.98}{\left(={1}-{0.02}\right)}\).

Therefore,

\(\displaystyle{X}\sim{B}{\left({20},{0.02}\right)}\)

Step 2

(b) Obtain the expected number of audits in a 20 year period.

The expected number of audits in a 20 year period is obtained below as follows:

The required value is,

\(\displaystyle{E}{\left({x}\right)}={n}{p}\)

\(\displaystyle={20}\times{0.02}\)

\(\displaystyle={0.40}\)

The expected number of audits in a 20 year period is 0.40.

Step 3

(c) Obtain the probability that a person is not audited at all.

The probability that a person is not audited at all is obtained below as follows:

The required probability

Use Excel to obtain the probability value for x equals 0.

Follow the instruction to obtain the P-value:

1. Open EXCEL

2.Go to Formula bar.

3. In formula bar enter the function as“=BINOMDIST”

4. Enter the number of success as 0.

5. Enter the Trails as 20

6. Enter the probability as 0.02

7. Enter the cumulative as False.

8. Click enter

EXCEL output:

From the Excel output, the P-value is 0.6676

The probability that a person is not audited at all is 0.6676.