Occasional reflections on Life, the World, and Mathematics

Our distant descendants hunkering in their radiation-proof underwater bunkers will speak of “Donald Trump’s lawyer” proverbially, as an oxymoronic self-flagellating professions, the way we might speak of “Bernie Madoff’s accountant” or “Jeffrey Dahmer’s nutritionist” or “Water-safety officer on the Titanic”. Tom Lehrer spoke on one of his satirical LPs from the 1960s about people following the news with unease, feeling “like a Christian Scientist with appendicitis”. One might similarly say “I feel like Donald Trump’s lawyer”.

Our story to date: When last we saw Sheri Dillon it was a week before Donald Trump’s inauguration, and she spoke beside a table full of binders — none of which were ever seen by the public — which supposedly showed that Trump was taking some unspecified action that would resolve all legal and ethical conflicts arising from his business interests. They were the most prominent unseen-document-political-props since Joseph McCarthy’s infamous “list” of “known communists”. (Or perhaps Mitt Romney’s “binders full of women”.)

Now she and her partner have produced a letter exonerating Donald Trump from illicit dealings with Russia. Except, not really. Accept for a moment that the word of ones own lawyer were probitive in such matters — and wouldn’t trials be much simpler if that were the case (“I have looked at my client’s records, and determined that he did not actually commit that burglary.” Very good, then, off you go. Sorry to have troubled you.) The letter (which is dated two months ago) very explicitly confines itself to evidence contained in Trump’s personal tax returns from the past 10 years, and merely says that he has had no direct income or investment from “Russian entities”. It does not touch on the possibility that there is income or investment from, say, Ukrainian entities, or investment through middlemen, or joint proprietorships, or former-Soviet business or government entities exerting influence on his behalf, or on behalf of his cronies. And any evidence of Russian collusion in offshore investments — so that they would be shielded from US taxes — or those that occurred more than 10 years ago, are likewise excluded. It also concludes by saying there are some exceptions, not detailed, that are “immaterial”, which is a strange word to use in a legal document. Does this mean a group of Russian tourists had lunch in the restaurant at Trump Tower? Or Russian oligarchs buying $10 million apartments? Never mind, it’s “immaterial”.

It reminded me in a funny way of the Monty Hall problem. As a reminder, this involves a game with a prize behind one of three identical doors. (The story is sometimes told with goats behind some doors and a car behind one, but as someone with no driving license but with a vague affection for medium-sized ungulates I always found that confusing.) You pick a door, let’s say door 2. Your opponent, who knows where the prize is, will then open a door — let us say door 1 — with no prize, and then offer you the possibility of choosing to keep door 2 or to switch to door 3. Under the assumption that your choice is made at random, and the opponent will always have to show you an empty door, you have a 2/3 chance if you switch, but only 1/3 chance if you keep your door. The interesting thing about the problem is not the mathematics, but that it conflicts strongly with some people’s intuitions.

Many years ago I made considerable effort to explain this intuitively to a group of nonmathematicians who simply refused to accept that this could be intuitively correct. I invented a number of variants that attempted to highlight features of the argument. In one, there are 100 doors. You choose one and the opponent shows you 98 empty doors, before offering you the opportunity to keep the one you chose originally, or switch to the one other remaining closed door. Imagine that you have chosen door #1. The opponent starts opening doors: 2, 3, 4,… He gets to #37, opens it, then skips to #39, and keeps on going up to 100. What seems more likely: That you picked the right door to begin with, and the opponent (who could then skip any door, since they’re all empty) chose #38 at random; or that he skipped #38 because he had no choice, because the prize was there? Most people intuitively feel like skipping #38 — the shadow of his actions, since it is only conspicuous as a negative image, by the action that was not taken — strongly suggests that there is some reason, and that the prize is highly likely to be there.

Similarly, when a lawyer is carefully detailing answers to some questions, and claiming to have reviewed some evidence to disprove some accusations, particularly when some are accusations that no one had actually made, it highlights the evidence and accusations that have not been addressed.

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