Probability that two specific people are selected for IRS audit within 4 year period A recent NYTimes article states that

The minuscule chances of the two highest-ranking F.B.I. officials — who made some of the most politically consequential law enforcement decisions in a generation — being randomly subjected to a detailed scrub of their tax returns a few years after leaving their posts presents extraordinary questions.


Was it sheer coincidence that two close associates would randomly come under the scrutiny of the same audit program within two years of each other?

According to the article, the probability that a given person is selected for a detailed audit in a given year is $p \approx \frac{1}{19250}$. (In reality this probability might vary slightly by year, but for the sake of simplicity let's assume it equals $p$ in every year.)
The article states that the two individuals in question were audited in different years, but within the same 4 year period of interest (coinciding with a presidential term).
What is the probability of the event "both of these people of interest are selected for an audit within a specified 4 year period"?
My first pass at an answer: the probability that a given person is not selected for an audit within a given 4 year period is $(1-p)^4$, assuming independence over time, and the probability of being selected for an audit once or more within the 4 year period is therefore $1 - (1-p)^4$. Assuming independence between individuals, the probability that both individuals are audited once or more within the 4 year period is therefore
$$
(1 - (1 - p)^4)^2 \approx \frac{43}{10^9}
$$
or approximately 43 out of one billion.
Is that answer (approximately) correct?
The article goes on to ask whether "something in their returns increase[d] the chances of their being selected" for the audit. If the IRS's audit selection process isn't a simple random sample of individuals/tax returns, but instead varies with observable variables like income (perhaps high-income individuals are more likely to be selected for an audit), could that meaningfully change the answer (for example, by changing the answer to something larger than $0.001$)?
Edit: anyone reading this will likely be interested in this companion article.

Your calculation of the probability $q$ of them being selected in the four-year period is correct.
And you are also right that this could be increased if the probability of being selected would be increased from $p = \frac{1}{19250}$ to something higher according to reasons like being suspicious or high income.
How high would $p$ have to be to get a $q>0.001$?
$$
\begin{align}
q &= (1 - (1-p)^4)^2\\
\leftrightarrow \sqrt q &= 1 - (1-p)^4\\
\leftrightarrow 1-\sqrt q &= (1-p)^4\\
\leftrightarrow \left(1-\sqrt q\right)^{1/4} &= 1-p\\
\leftrightarrow p &= 1 - \left(1-\sqrt q\right)^{1/4}.
\end{align}
$$
Thus, for $q = 0.001$, we get:
$$
\begin{align}
p &= 1 - \left(1 - \sqrt{0.001} \right)^{1/4}\\
  &\approx 0.008.
\end{align}
$$
I.e. if the selection probability of an individual for audit in a given year would increase above 0.008, then the described event could happen with a probability higher than 0.001.
