The Monty Hall Problem

 “It’s easier to fool people than convince them they’ve been fooled.” – tenuously attributed to Mark Twain

Anchor bias: the tendency in decision-making and related cognitive processes to unduly emphasize information obtained at a certain point in time over any newly-discovered information. This bias tends to take the form of sticking with an initial position, and is likely informed by avoiding the mental effort and presumed risk of changing one’s position from that which is currently held.

Or something like that.

Note, much of the background information for today’s column was derived from a Wikipedia article on the subject. You could read that, but this dispatch is way funner!

The Monty Hall Problem was first posed in American Statistician by one Steve Selvin in 1975, and later in a 1990 magazine column by Dr. Marilyn vos Savant, great-granddaughter of the famed idiot Cletus Savant. JK about that last part, but she is descended from noted 19^th and early 20^th century physicist Ernst Mach. Side note – you’ve really made it in science if your name is used in lowercase, i.e., a unit of measure is named after you. In this case, mach is the ratio of an object’s speed relative the speed of sound.

Marilyn vos Savant was, among other things, a rather brainy columnist in Parade Magazine.  Those of us with a degree of superannuation will remember the roto gravure magazines and ad flyers that accompanied the colored Sunday comics in the rainforest-killing newspapers of yesteryear. Parade Magazine was one such, and in it, vos Savant wrote “Ask Marilyn,” an intriguing Q&A column on brainy topics.

On September 9, 1990, vos Savant posed her most controversial column, on the Monty Hall Problem. It’s not the problem itself that’s controversial; it’s based on the (at the time) popular game show Let’s Make a Deal, a show with which most of her readers would be familiar, with only 3 or 4 VHF TV channels to choose from, and even worse pickings on UHF. The controversial part was the solution, and the rigorous and often contentious debate that followed.

First, let’s lay out the problem. The game show contestant is presented with three closed doors: A, B, and C. The contestant is informed that behind one door is A NEW CAR! Behind the others are goats.

A few assumptions are in order at this point:

1) The doors all look alike, are the same size, etc., i.e., not a garage door and two goat pen gates.

2) The car is not a Rivian and is therefore preferable to a goat.

3) The contestant is sufficiently urbane as to prefer the car over a goat.

4) The goats remain quiet and do not emit any telling odors.

And some game rules to remove any bias or other chicanery:

1) The a priori selection of which door conceals the car is assigned at random with a uniform probability distribution, i.e., it’s completely unbiased.

2) The assignment of door to car or goat does not change throughout the game, although the goats may discreetly trade places with each other.

3) Which door conceals which prize I s known to the host.

4) …and is not known to the contestant.

So much for the setup, now the game:

The contestant chooses a door, hoping it conceals a car. Monty Hall does not immediately open the door. Instead, he opens another door, revealing a goat. Remember, Monty knows where the goats and the car are. At this point, Monty offers the contestant the opportunity to change his or her initial choice to the other closed door.

The contestant then opts to stick with the original door choice or switch. Monty then dramatically opens the other doors to reveal the contestant's fate, car or goat, euphoria or despair, that's show biz! And speaking of show biz, I have no idea whether or not any contestant actually kept the goat. I know I would, and my next stop would be a halal butcher, but that’s beside the point.

So... what’s the problem?

The problem is, should the contestant stick with the initial choice, or, when given the opportunity, choose the other unopened door? Answer: switch!

vos Savant explained why in her famous column, and the controversy was that most of her readers, including some real brainiacs, didn’t believe it. That’s because, although we encounter hands-on applications of it on a daily, if not hourly basis, most people are not comfortable thinking consciously in probability space. But if today’s discussion goes well, you will be, at least more so than you are now.

I’m sure you can appreciate that the initial choice has a 1/3 probability of being correct. Therefore, the car has a 2/3 probability of being behind one of the unchosen doors. Then Monty supplies more information by revealing which of the unchosen doors conceals a goat. Maybe both unchosen doors conceal goats, or maybe just one did; in any case the contestant now sees one of the “wrong” choices.

So, of what value is this information? Note that information was added about the non-chosen doors, not the chosen one. The chosen door still has a 1/3 probability of being correct, but the new information assigns the 2/3 probability all to the unchosen but still closed door. 1/3 odds of winning by staying with the initial choice, 2/3 by switching. Therefore, switch!

Get it?

No? Consider this: instead of three doors, the contestant has 1000 from which to choose. Let’s say for simplicity that he or she picks door #1, and we can all agree that the odds of winning the car are 1/1000. The odds of the car being behind one of the unchosen doors is 999/1000, i.e., a near-certainty. Next, Monty opens up 998 doors revealing goats (the studio must be getting rather redolent by this time, and probably a bit noisy as well) – remaining closed are door #1 and door #637. Now, you and the contestant can see, of all the bad choices available earlier, only one (of the two closed doors) remains, and he or she would be exceedingly lucky not to have made one initially.

vos Savant’s column generated over 10,000 letters (that’s how people flamed before the internet), with only 8% of the respondents agreeing with her. Note, this does not mean that only 8% of readers agreed; probably more did, but would be less inspired to lick a stamp to make their case. The debate raged on for many months, in and out of the “Ask Marilyn” column, among academics and laypeople alike, accompanied with much acrimony and a side order of sexism. Numerous proofs and explanations failed to move the needle, and the controversy was ultimately resolved by schoolteachers across the US volunteering their class time to conduct simulations with their students (crowdsourcing before crowdsourcing was cool), empirically proving the point. In the end, only 70% (up from the initial 35%) of academics agreed with her. Not stated was what kind of academics – there may have been some Harvard faculty in the pool.

For today's little statistics and probability lesson, we'll let Clint Eastwood have the final word.