The Manhattan U.S. attorney has filed criminal charges against SAC Capital Advisors LP, and the Securities and Exchange Commission has brought an administrative action against the hedge fund’s founder and chairman, Steven A. Cohen. Both actions are based, in part, on allegations that Cohen read an e-mail warning him of soon-to-be announced disappointing earnings at Dell Inc. that allowed him to sell his position and avoid a $1.7 million loss.

In a white paper responding to the SEC action, Cohen’s lawyers said he received about 1,000 e-mails a day during the period in question, and that he opened only 21 percent of the messages sent to him by the employee who sent the Dell e-mail. One might assume, then, that there’s a 21 percent probability that Cohen used knowledge from the e-mail to trade. That amounts to odds of about 4:1 against Cohen having read the crucial e-mail.

At first glance, that seems pretty good for Cohen. Yet Bayes’ rule, a statistical theorem that can be used in legal proceedings to calculate the probability of an event, complicates the picture and provides a clue to the strategy Cohen’s lawyers might deploy to defend his case.

The real question about the e-mail, of course, isn’t just whether Cohen read it, but whether he did so before he traded the Dell stock. We know that Cohen sold his position about 10 minutes after he received the e-mail. What we don’t know is whether he read the e-mail before trading, after trading or not at all.

Even Odds

Let’s assume for the time being (though Cohen’s lawyers dispute this assertion) that the e-mail contained material, nonpublic information that, had he been aware of it, would have prevented Cohen from trading lawfully. Now let’s compare the probability that Cohen read the e-mail before he traded the stocks and the probability that Cohen didn’t read the e-mail before he traded them.

Bayes’ rule tells us that the probability that Cohen traded after reading the e-mail is equal to the product of the probability that he traded given that he read the e-mail before trading and the probability that he read the e-mail before trading. The probability that Cohen traded without relying on the e-mail is the sum of two probabilities: the probability that he traded and never read the e-mail plus the probability that he traded before reading the e-mail (which wouldn’t be illegal, as it wouldn’t have affected his trade). All these probabilities focus on the likelihood of trading, not just the likelihood of reading the e-mail.

This is where the exercise becomes more interesting, because the probability of trading is likely dependent on whether Cohen read the e-mail. That means the case for the prosecutors may focus less on Cohen’s e-mail reading habits and much more on the coincidence of the e-mail’s appearance in his inbox (an undisputed fact, so far as we know) and the Dell trade 10 minutes later.

Let’s assume the prosecutors argue that the e-mail warning of weak Dell earnings was almost sure to prompt Cohen to sell his position. In terms of probability, let’s suppose this means there was a 95 percent probability that Cohen would sell his Dell position if he read the e-mail before his trade. That means 19:1 odds that Cohen would trade if he read the e-mail beforehand.

Suppose, however, that Cohen’s lawyers can demonstrate that he had other good reasons to sell his Dell position at the moment he did, even if he didn’t read the e-mail beforehand. To put a number on it, let’s assume that given the other information Cohen had (such as the trades that day of one of his favorite employees who also was selling Dell), there was a 50 percent probability that Cohen would sell Dell. In other words, there were even odds that Cohen was going to sell his Dell that day no matter what.

Crunching Numbers

Now let’s also assume that there was a 15 percent chance (5.6:1 odds against) that Cohen read the e-mail before trading, an 80 percent chance that he never read the e-mail (4:1 odds against), and a 5 percent chance (19:1 odds against) that he read the e-mail later (after all, only about 10 minutes passed between the e-mail and the trade, and there is evidence that Cohen was otherwise engaged for much of that time).

These suppositions allow us to use Bayes’ rule to calculate that the odds are about 3:1 that Cohen didn’t read the e-mail before he traded. Again, good odds for the hedge-fund mogul.

Of course, the answer depends on the numbers one chooses to crunch -- that is, the assumed probabilities. To illustrate how the calculation would change with different probabilities, consider what would happen if -- and there is no evidence of this -- Cohen read 50 percent of the e-mails from that particular employee, and 40 percent of those within 10 minutes of receiving them. And what if the information available to Cohen when he traded the Dell stock created only a 25 percent chance of his selling Dell at that time? With these assumptions, the odds are more than 2.5:1 that Cohen did read the e-mail before he traded. Now, not such good odds for Cohen.

Only Cohen knows whether he read the e-mail before he traded the stock. His lawyers are smart. The white paper shows they are focused on a number of probabilities that matter: the probability that Cohen read the e-mail at all and the probability that he had other reasons to trade. And Cohen’s lawyers also argue that the e-mail didn’t contain inside information to begin with. The ever-handy Bayes’ rule may help determine the outcome.

(J.B. Heaton is a partner at the law firm Bartlit Beck Herman Palenchar & Scott LLP in Chicago. Nicholas G. Polson is professor of econometrics and statistics at the University of Chicago Booth School of Business.)

To contact the writers of this article: J.B. Heaton at jb.heaton@bartlit-beck.com; Nicholas Polson at ngp@chicagobooth.edu.

To contact the editor responsible for this article: Max Berley at mberley@bloomberg.net.