Trump’s Monty Hall Problem
How The Odds Might Change That GOP Senators Will Flip
Sometime in the next 12 months, Donald Trump will almost certainly be impeached. But what is the likelihood that at least 67 senators will vote to convict? More importantly, how will this probability change over time as revelations of Trump’s criminal conduct continue to spill into the public sphere? In December, famed Watergate journalist Elizabeth Drew suggested impeachment was “inescapable” — that public pressure will swell as time goes on. “Too many people think in terms of stasis: How things are is how they will remain. They don’t take into account that opinion moves with events.”
To better predict how the steady stream of criminal revelations might change the political calculus of GOP senators, the famous Monty Hall problem is helpful. This counter-intuitive probability puzzle has a surprisingly simple solution, but only if the problem solver recognizes a critical insight into how odds change as new information emerges.
If the Democratic-controlled House impeaches Trump, conviction and removal would require a two-thirds Senate majority, or 67 senators. Assuming all 47 Democrats vote to convict, at least 20 Republicans will need to join them.
It likely will not be linear, but the odds of Senate conviction will increase as the public is exposed to hard evidence of Trump’s criminal conduct, obstruction of justice, and abuse of power. Investigators are working their way to the very top of the Trump syndicate, and special counsel Robert Mueller is currently batting 1.000. So far, he has indicted or secured guilty pleas from at least 34 individuals and three companies. Many of these individuals are cooperating with investigators. Federal prosecutors in other jurisdictions are also closing in on Trump.
At its core, the Monty Hall problem is about improving one’s gamble, given two options, after receiving subtle but critical information.
Some background: The Monty Hall riddle was inspired by a popular game show in the 1960s and 70s called Let’s Make A Deal. On the show, the handsome and charming host, Monty Hall, asked each contestant to choose one of three doors. Behind one door was a car; behind the other two was something silly nobody wanted, like a goat. Naturally, it was the goal of every contestant to win the car. Contestants were free to choose any of the three doors, but before the door was opened, Monty would do his thing.
Unlike the contestants, Monty knew which door was the winner. So, he would open one of the other doors, and reveal a goat. He would never open the door with the car behind it; that would ruin the game. And regardless of which door contained the car, Monty would never begin by opening the door the contestant had picked initially.
After opening one of the doors and revealing a goat, Monty would give the contestants a choice: stick with their first guess, or switch to the remaining mystery door. Viewers probably took a certain twisted pleasure as contestants agonized over whether they should take Monty’s offer and switch their pick, or stick with their gut — their first choice. Ahh! What to do?!
The Monty Hall problem became more than just the basis for a cheesy game show when, in September 1990, it suddenly sparked a fierce debate among leading experts in mathematics and probability theory. In truth, the name of the “Monty Hall problem” had actually been conceived in 1975 by statistician Steve Selvin, who used the moniker in a letter published in the scientific journal The American Statistician, titled “A Problem in Probability.”
But the problem remained relatively unknown until controversy erupted 15 years later, when advice columnist Marilyn vos Savant answered a reader’s question about whether it was better to always take Monty’s offer and switch doors, or to stick with the first pick. The “Ask Marilyn” column was featured in Parade magazine, and together with syndicated newspapers, it reached a combined circulation of almost 35 million people. Marilyn’s qualification for writing it was that she had long held the Guinness World Record for the highest IQ ever recorded (228).
In his book, The Drunkard’s Walk: How Randomness Rules Our Lives, Leonard Mlodinow recounts how this obscure riddle captivated so much attention and became the show’s principal legacy:
This issue has immortalized both Marilyn and Let’s Make a Deal because of the vehemence with which Marilyn vos Savant’s readers responded to the column. After all, it appears to be a pretty silly question. Two doors are available — open one and you win; open the other and you lose — so it seems self-evident that whether you change your choice or not, your chances of winning are 50/50. What could be simpler? The thing is, Marilyn said in her column that it is better to switch.
Marilyn’s answer was met with anger and scorn. Approximately 10,000 readers — nearly 1,000 of them with PhDs — wrote letters to the columnist. Many were esteemed mathematicians who felt the need to “mansplain” the error of Marilyn’s ways. One math professor from George Mason University condescended, “You blew it . . . As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.” Even Paul Erdős, one of the most prolific mathematicians of the 20th century, grew visibly upset with colleagues and refused to accept Marilyn’s explanation until he was shown a computer simulation that demonstrated the result.
Marilyn was right, of course; contestants who switch doors double their probability of winning the car compared to contestants who stick with their initial choice. But how can something seemingly so obvious (a 50/50 probability regardless of which door is ultimately chosen) be the wrong answer, and trip up so many experts — including the greatest mathematician of his era? The answer lies in their failure to recognize a critical insight: When Monty opens one of the doors and reveals a goat, he is not choosing that door at random. For this reason, Monty’s choice provides key information to the contestant.
If you’re still struggling to understand why it is always better to switch, this short explainer video is helpful. But here is the simplest explanation: When the contestant first chooses — at random — one of three doors (let’s call this the “chosen” door), each door has a ⅓ probability of containing the car. So the chosen door initially holds ⅓ probability, and the combined probability of the other two doors is ⅓ + ⅓ = ⅔. But, because Monty knows which door contains the car (and will never open that door), and because he won’t open the chosen door, the door he does open subtly reveals a key nugget of new information the contestant can — and should — use. The moment Monty opens that door and reveals the goat, all of the ⅔ probability from those two doors gets compressed into that single, untouched door. Thus, contestants are left with the ⅓ probability that their chosen door is the winner, compared to the ⅔ probability that the untouched door contains the car. Which is to say, they literally double their chances of winning by switching doors. Computer simulations, as Erdős came to realize, bear this out.
Getting back to Trump, congressional Republicans have made their initial choice by blindly defending the head of their party and refusing to provide congressional oversight. Some members have actively helped Trump cover up his crimes. “The problem for Republicans,” Brian Beutler points out, “is that they have no idea what crimes they’re preparing themselves to absolve.”
Right now, GOP senators calculate it is politically safer to continue to protect Trump than to flip on him, lest they face the wrath of Trump’s legions in Republican primaries. But their current position — that Trump is clean because Mueller has not yet indicted any Americans in the Russian conspiracy — is increasingly untenable.
Ironclad evidence of Trump’s treachery would shift that calculus. As more and more proof emerges, Republican senators may soon realize that switching doors is the wiser gamble. Previously, I wrote that despite Trump’s current popularity among Republicans, the interplay between his criminal conduct, the American judicial system, and his political standing will be more dynamic than many appreciate. Indeed, Americans are quickly learning the extent of Trump’s life of crime and outright corruption.
GOP senators who flip on Trump might be few at first. But already there are signs of pressure building behind the dam. In fact, several senators have bucked Trump on key issues, most prominently funding for his half-baked border wall and a national emergency declaration, but also a report on the murder of Saudi journalist Jamal Khashoggi, Saudi Arabia’s war in Yemen, and Syria and Afghan policy.
The Monty Hall problem is a brainteaser that illustrates how the odds of “guessing” correctly between two options improve significantly when the contestant learns new information partway through the game. Sometimes, the new information changes the odds in a way that is surprising or even counter-intuitive. Like contestants on Let’s Make A Deal, GOP senators can’t know with certainty which choice — protect Trump at all costs, or throw him overboard if things get bad enough — will turn out to be the correct political judgment. But the addition of devastating evidence of Trump’s criminality and abuse of power just might change the odds in a way even veteran political observers don’t anticipate.
Those who discount the possibility of Trump’s expulsion from office compare the current era to Watergate. If Richard Nixon had enjoyed Fox News and today’s hyper-polarized electorate, they argue, he would have survived. That’s probably true, but it misses a key point: Trump’s criminal conduct is far worse than Nixon’s, in both scope and impact. The American people will be the ultimate judges of that, but when the time comes, Republican senators would do well to consider how revelations of Trump’s crimes affect the odds they’ll guess right.
To be sure, evidence of Trump’s brazen self-dealing and obstruction of justice will test Republican senators’ commitment to putting their own naked political interests ahead of the rule of law. But among GOP senators who still care about U. S. national security, revelations that lay bare the extent of Trump’s outright betrayal of American national interests may well be the sins that spark his political collapse.
In arguing with Marilyn vos Savant, the mathematician Erdős long refused to accept he was wrong. Only when confronted with hard evidence from a computer simulation did he acknowledge the reality staring him in the face.
As public opinion changes in response to forthcoming explosive revelations, how many Republican senators will decide to switch doors and flip on Donald Trump? Perhaps it is fitting that America’s game show host president may soon be facing a serious Monty Hall problem.