Sunday, August 02, 2009

The Monty Hall Problem Meets The Bachelorette

Last Monday evening inexplicably found the Holy Prepuce attending a Bachelorette viewing party. In a turn of events that almost merited its billing as "the most dramatic Final Rose Ceremony ever," previously-eliminated bachelor Reid Rosenthal returned unexpectedly to the program.

Reid's arrival greatly surprised bachelorette Jillian Harris, who had just dispatched finalist Kiptyn Locke and was about to declare her love for alpha male Ed Swiderski. (In the post-feminist world of reality television, the distaff may choose its betrothed, even sampling candidates' sexual talents along the way, but the marriage proposal must come from the spear.)

My fellow viewers expressed differing views as to whether Jillian should dump Ed for Reid. I had no opinion as to the gentlemen's relative merits, having seen too few episodes to form one, and was disappointed at my inability to participate in the exercise.

But then, I had a brainwave: Jillian must switch. Why? The Monty Hall Problem, of course.

Imagine that Jillian is a contestant on Let's Make a Deal. The host, Monty Hall, shows her three doors, and tells her there is car behind one of them. Behind the other two are goats. (As a child, I often wondered whether losing Let's Make a Deal contestants were required to take home the livestock, and if so, whether they took proper care of it.)

Jillian chooses door #1. Monty, who knows where the car is, opens door #3, revealing a goat. Monty now gives Jillian the options of standing by her selection of door #1 or switching to door #2. Assume that under the rules of the show, Monty must always open an unpicked door, must only reveal a goat (not the car) in doing so, and, regardless of where the car is, must always offer the opportunity to switch doors. Assume also that when Monty has a choice of doors to open--i.e. when both unpicked doors contain goats--he is required to choose at random.

The question: should Jillian stay with door #1, should she switch to door #2, or does it make no difference? The answer, which nearly everyone gets wrong, is that she should switch to door #2. Doing so doubles her chance of winning the car.

People usually get this wrong because they remain fixated on the one-in-three a priori odds of the car being behind any particular door. Opening a door doesn't change where the car is, they reason, so the relative odds of it being behind #1 or #2 remain what they were: even.

The conclusion that switching makes no difference fails to recognize the conditional probabilities that kick in with the new information conveyed by Monty's choice. Formal explanations are discussed here, but put simply, where a contestant has chosen door #1, Monty has only a 50/50 likelihood of opening door #3 if the car is behind door #1, but must open door #3 if the car is behind door #2. Hence, Monty's choice to open door #3 is twice as likely to signify a car behind door #2 as it is to signify a car behind door #1.

On The Bachelorette Jillian isn't trying to pick a car, she's trying to pick the perfect husband. But unlike in normal life, where she would have the opportunity to choose from many men encountered over a number of years, her choice on The Bachelorette is subject to Let's Make a Deal-like constraints. A contestant on Let's Make a Deal can't go hunting for free cars wherever and whenever she wants, she must take her chances right now, with the three doors before her.

For the sake of analysis, let's take the final two Bachelorette episodes as a single "game." Thus, we'll treat Jillian's decisions to eliminate Reid and Kiptyn as a single event, i.e., the selection of Ed out of three possible suitors.

We're allowed to do this because of my next assumption, which is that Jillian's conscious mind has become confused. It did an OK job of whittling the field of bachelors down to three, but the exhausting whirlwind of travel, green-room liquor, and sudden stardom has negated its ability to choose intelligently among the finalists. Jillian's selection of Ed is based on irrelevant heuristics, as was the order in which she eliminated Reid and Kiptyn.

Assume also that Jillian's subconscious mind is unaffected; that it wants what's best for Jillian and knows which one of the three men is right for her. But being subconscious, it can't fully direct Jillian's decisions. It can only cause her to consider, or not consider, certain courses of action.

Until Reid shows up, Jillian's conscious mind has been driving her decisions. The rules of The Bachelorette require that she eliminate a prescribed number of bachelors in each formalized Rose Ceremony, and she has dutifully complied. But with Reid's return at the emotional climax of the series, the producers have changed the rules unexpectedly. Jillian's guard is down, and her subconscious mind seizes the opportunity: it makes Jillian consider taking Reid back.

Jillian's subconscious mind is Monty Hall. Jillian thought she had made a one-time choice (Ed) among three "doors." Her subconscious has presented her with a second chance at one of the "doors" (Reid) after exposing the third "door" (Kiptyn) as a "goat."

How do we know that Kiptyn is a goat? Let's make another assumption here: out of fairness, the same Bachelorette producers who changed the rules to permit Jillian a second chance at Reid would also allow her a second chance at Kiptyn. In other words, when Reid walked onto the set, Jillian could have said, "I know I don't want Reid, so if I get a do-over, I'll think about using it on Kiptyn." (Given the Molly / Melissa switch on the last Bachelor, the assumption is not unreasonable.) Jillian's subconscious thus has the power to "open" either Reid or Kiptyn by causing Jillian not to reconsider him. And just as Monty Hall is not permitted to eliminate the car, presumably Jillian's subconscious isn't going to "open" the perfect husband and subject Jillian to the pointless cruelty of choosing between goats. Jillian doesn't think about a Kiptyn do-over; Kiptyn is "opened"; ergo Kiptyn is a goat.

A final assumption: Reid's reappearance is sufficiently destabilizing that it compels Jillian at least to consider dumping Ed. Jillian's subconscious can't force her to think of no one but Ed, even if it believes that he is the perfect husband. Jillian's conscious mind must make the choice between Ed and Reid.

So, should Jillian switch to Reid? The Monty Hall Problem tells us yes. Given Jillian's initial choice of Ed and Reid's return, it is two to one that the "opening" of Kiptyn signifies that Reid is the "car"--the perfect husband.

Unfortunately (SPOILER ALERT, in case the finale is still queued on your TiVo), Jillian does not switch, and is now engaged to Ed. She has settled for an ill-suited marriage, life's ultimate game of Let's Make a Deal.

This has been the nerdiest Rose Ceremony ever.

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