Will the GOP Senators stick with door number 1 or will they switch to door number 2? @RyanCaseyWA

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).
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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.

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.