The New York Times reported that the Internal Revenue Service conducted one of its most rigorous types of audits with former FBI Director James B. Comey and his former Deputy Chief Cabinet Secretary Andrew G. McCabe.
This prompted many perfectly rational questions, most of them are the following variants: what are the odds? As the article pointed out, it is highly unlikely that President Donald J. Trump’s two high-ranking political enemies were audited by pure coincidence.
However, the minimum is not zero.
If you want to believe this is a coincidence, how unlikely do you think it is? Here we try to estimate that probability as seriously as possible.
First, the fact: Both men were selected for audit under the National Research Program (NRP), which is a small part of all audits conducted by the IRS each year. These audits scrutinize a sample of returns to collect data on tax compliance.
According to the IRS, there were approximately 5,000 such audits in 2017, 4,000 in 2018, and 8,000 in 2019, selected from approximately 154 million tax returns each year. Mr. Comey’s audit was against the 2017 tax return. McCabe was for his return in 2019.
Many aspects of the NRP complicate calculations, such as the sampling method of the IRS auditor and the various years of the audit itself. We will return to these issues later. For now, all taxpayers are equally likely to be audited, and we assume that both men were audited in 2017.
If this problem is mentioned in a textbook on probability, then:
If you have 154 million marbles (approximate number declared each year) in a giant jar, some of which are red (representing Mr. Comey and Mr. McCabe), what are your possibilities? If you randomly subtract thousands (the number of audits for the year) from the jar, will you get more than one red marble?
It may sound complicated, but Relatively well-studied problemsSomething that many math or statistics majors encounter in college classes. People have already derived equations for estimating these probabilities. Hypergeometric distributionThere are applications such as election audits and card counting.
Simply enter an estimate of the total number of marbles, the number of red marbles, and the number of draws to get the odds.If you have only two red marbles, that is, if you limit your exercises to: that’s all McCabe and Comey — This equation gives a probability of about 1 in 950 million.
These are odds that are much steeper than your chances of winning Powerball. Also, the result is almost meaningless. At best, it’s the correct answer to the wrong question.
To understand why we need to acknowledge the absurdity inherent in our movement: the fact that we know it has already happened to best estimate the likelihood of an unlikely event. Must be set aside. (The probability that it happened is 100%.)
Jordan Elenberg, A professor at the University of Wisconsin, who wrote a book on mathematics and reasoning, explains: “In one counterfactual universe, what is the probability of this happening already happening in our universe?”
It may seem strange, but the same problem arises with probabilistic exercises that are as basic as a coin toss.
If you flip the coin 20 times, the particular sequence of heads and tails is very rare, about one millionth, but it actually happened.When A few The flip sequence always happens. It’s a surprising coincidence only if it’s the sequence you tried to get before playing.
Similarly, it would be a mistake to narrow your search to just Comey and McCabe. This is because if you know the two, you are likely to look at these probabilities. other On behalf of these two men, the administration’s notable political enemies were audited.
Better questions are: How likely are you to have more than one person? favorite Will Comey and McCabe be audited during this period?
Do people in this group need to include the two top FBI executives? Are there two top executives in the Justice Department? It is this framing (a subjective decision, not a fact) that determines the probability estimation more than the statistical distribution or the choice of sampling weights.
This ranges from two (Commy and McCabe and no one else) to 400 (a conservative estimate of the number of American Trump), with a variety of choices in the number of red marbles of ours. It is a graph of the probability that the equation brings. Since running for president, he has been insulted by name on Twitter).
Choosing who should be considered red marble alongside Comey and McCabe will dramatically increase your odds.
The important thing is not to decide the number, but to recognize that the choice of group size determines our answer. Some speculations are certainly better than others, but many options are defensive.
Correspondence to details
Now let’s narrow down the more realistic ones and go back to some of the things we ignored in a simple interpretation of this problem.
First, the two men were not audited in the same year. By expanding the scope to cover the three years from 2017 to 2019, the resulting probability is greatly increased. This is easy. If you have some chance of being audited in a particular year, it means that the more years you have, the more opportunities you have to be audited.
Second, I’m only interested in the probability that at least two people will be chosen.. The possibility that the same person will be chosen twice is not considered. According to Comey, it’s unlikely that the audit will last for more than a year. Note that we are looking at the probability that at least two people will be selected, not exactly two, as it will also be important if three or more individuals are selected from the group.
Finally, the IRS does not select people in a truly random way. Instead, agencies tend to choose several types of taxpayers, including: High-income earners, More often than others. For the 2001 tax year, the NRP sample Return value From people in the 90th percentile of income, it was about 1.7 times the expected rate and was the income of choice separately from income. That rate soared above the highest income rank, so those with the top 0.5% of income could be included in the sample more than 10 times more than those near the median income.
We can probably speculate that any group of Mr. Trump’s enemies will earn more than a random sample of Americans. However, it is not possible to realistically estimate the full annual income of everyone in our group. We also know that the IRS considers other factors in sampling. Types of returns The taxpayer files and its sampling method may change from year to year. This provides little guidance on how to match the IRS methods. Therefore, we leave an estimate that is not weighted by income. As a calculation on the back of the envelope, if you are worried about how income will affect these results, double the probability of the result if you think the income of the members of the group is very high, if you think they are Can be multiplied by 10. ‘I’m very rich.
Put them all together
Incorporating these options, the table below shows the estimated probabilities depending on the size of the group under consideration...
Or, if you’re not good at making choices, you’ve created a simple calculator to create your own probabilities.
So which estimate is “correct”?
The most realistic output of this equation can be explained exactly “very rare” or even “very rare”, but there is no evidence of cheating.
“It’s a bit like an irresistible force and an immovable object,” he said. Andrew German, Professor of Statistics and Political Science at Columbia University, when spoken in summary about this exercise. “On the one hand, you say it’s completely random. On the other hand, it doesn’t seem to be.”
German, like all other statisticians who talked to the Times on this issue, said the biggest hurdle was not the details, but the definition of the question itself.
When trying to calculate the probability of a particular event because I think it may not be random, which puts me in a complicated position trying to imagine how I predicted the possibility of an event. Previous Said it happened David Spiegelhalter.. He leads the Winton Center for Risk and Evidence Communication at the University of Cambridge. The University of Cambridge is an organization dedicated to improving the way quantitative evidence is used in society.
Mathematics is easy, but it’s difficult to formulate a question. It is adjacent to “meaningless”, mainly because of how difficult it is for us to identify the group of interest.
“‘What is the possibility of this happening?’ It’s a simple statement,” he said. “This is a familiar statement, but in reality it’s a very difficult question to answer.”
There are limits to mathematics. According to German, the point in trying to estimate such a probability is not to put too much inventory in the numbers, and the result is to encourage you to know more.
In this case, the best question is not the answerable question you can look up in a statistics textbook.
Instead, German said the question raised was “what’s happening?”
Matthew Cullen Report that contributed.
