My young friend Victoria had been wanting to take me to this speakeasy she knew about downtown for the longest. We had our chance the other night to go so we went. Only thing is you can’t just show up, you have to be invited. So, Victoria went through the necessary machinations to secure our invite and we showed up on time in our finest costumes for the occasion.
We walked through the unprepossessing door from an alley off Main Street. We entered a small cubbyhole of a space manned by two standing gentlemen and a woman seated at a desk in front of a locked door which was located directly behind them.
We showed our IDs and it was dutifully checked against a roster resting on the desk. When our names were found the woman handed our ID’s back and slightly nodded to the gentleman guarding the door. He swung it wide and we stepped through to the top of a sharply declining stairwell.
As we made our way down the steep stairway, I couldn’t help but notice the atmosphere changing with each step. The air seemed to grow denser as if perfumed by some unknown censer. The lighting changed gradually and it seemed to give off a soft reddish glow. The temperature was getting colder by degrees the lower we went. At the bottom of the steps we were greeted by our smiling host. In the background we could hear music of the period playing and I swear I could hear the strains of “Put on a Happy Face.”
“Step this way please.”
We followed our host past a long and rather ornate wooden bar into the inner sanctum of Hell or High Water to our assigned seating. We arrived at a small round glass topped table flanked by two high backed leather chairs. He placed two drink menus in front of us and said, “Your server will be with you shortly.”
We looked the menu over and tried to decide what specialty cocktail to order. When the server came over, I decided to ask for a recommendation.
“Do you like the smell of smoke or leather?” he asked.
I allowed as I did.
“Well then, I recommend Sparks Fly.”
I took a look at that and saw it contained Mezcal, Cardamaro, Benedictine, Crème de Cacao, and Gun Powder Proof Rum. Sounded like an explosive concoction.
“Ok. I’ll try that.”
Victoria had the Devil’s Advocate, which was fitting.
I looked behind me at the room and on the back wall was a gigantic bookcase filled with books. The lighting was extremely dimmed and the music hushed.
As we sat sipping our drinks and soaking up the atmosphere, I was searching my head for something unfoolish to say. Victoria is such a good listener I wanted to come up with a good story that would put her in awe and elicit her rapt attention. She was my best audience.
I thought about a book that I had been reading and there was a particularly good scene in it I wanted to share with her about a logic problem. Victoria liked logic problems.
“I say, have you heard about the Monty Hall problem?”
She shook her pretty head no. Her eyes fairly glistened in the low light.
“Well, there’s this book I’m reading called, Sweet Tooth. It’s by Ian McEwan. Very clever piece about a female British spy in the 70s. In one of the chapters the protagonist, Serena Frome (rhymes with plume) and her lover/writer/friend Tom Haley were having dinner in their favorite seafood restaurant in Brighton and Tom says, ‘I’m always telling you stories about poems and novels but you never tell me anything about math. It’s time you did. Something counterintuitive, paradoxical.’
“Serena thought for a while.”
‘Well there was this one story making the rounds at Cambridge while I was there. It’s called the Monty Hall Problem.’
I took a sip of my drink and paused for emphasis.
“So, let me tell it to you as best as I remember it. I think you will like it.”
“It seems there are three boxes. Two are empty and one holds a fabulous prize, like an all-expense paid vacation to some exotic place on earth. You have to choose which one you think it might be in. You choose box number one. The host, Monty Hall, who knows what’s inside each box, opens another box. Say, box number three. It’s empty. He then says to you, ‘Do you want to choose box number two or stick with box number one?’
I then asked Victoria which she would do. She says it doesn’t make any difference because you have a fifty-fifty chance either way.
“Not true,” I say. “If you switch you have a two in three chance of winning. If you stick you only have a one in three.”
“No. that can’t be. If you have two boxes remaining, it’s a fifty-fifty chance.”
“I know that’s what it seems like, but if you do the math that’s not right. It’s sort of a paradox. It’s really about re-evaluating your decisions as you get new information. Monty filters your choice by opening one of the boxes. You now have new information. You know the fabulous vacation is not in box number three. This changes the odds.”
Victoria sat back in her leather chair and stared into the middle distance processing this information. I saw in her face the slow signs of recognition as she grappled with the problem and gradually came to understand the solution with the new information I had supplied to her.
“Oh! Now I get it. I don’t know why I didn’t at first.”
“That’s because it’s counterintuitive. Most people don’t at first. By the way. Tom didn’t get it at first either. Now here’s the kicker, getting back to the book. Tom takes this math problem and decides to incorporate it into one of his short stories.”
The server came over and asked us if we’d like another drink.
I nodded my assent and said, “Yes, but I think I’ll have something more traditional this time. Do you have Old Forster?”
He says, “Yes.”
“Good! Well then, I’ll have an Old Forster and soda. Club soda.”
He writes that down.
“And for the lady?”
Victoria says she’ll have a rum and coke.
The server gives a slight bow and disappears back into the gloaming.
“So,” I say. “Getting back to the book. Tom and Serena spend the rest of the weekend together back at Tom’s apartment. He claims to have had an epiphany and now totally understands the solution to the Monty Hall problem, although at first, he insisted just like you, that there was only a fifty-fifty chance the prize was in box number one or two. Serena gave him another way of looking at it. She said what if there were a million boxes? And you choose box six hundred thousand? Monty opens all the other boxes except box number ninety-seven. Now the only closed boxes are yours and ninety-seven. What are the odds now? Tom still insists fifty-fifty. ‘No! It’s a million to one against it being in your box. And an almost certainty it’s in the other! Finally, he gets it.
“So, they go back to the apartment and Tom thanks her for the idea and starts writing a story about the problem. He calls it, “The Adultery Probability.” They make love, eat left overs and on Sunday afternoon Tom escorts Serena to the train station. She takes the next train back to London.
“Monday morning, she is back at her job at MI-6. Tom doesn’t know she is a spy and is responsible for his new found fortune of being awarded a financial grant so that he doesn’t’ have to work and instead can concentrate on his writing. This is the “dirty little secret” that is hanging over Serena’s head and stands between them like the sword of Damocles as she tries to figure out how and when to tell him about it.
“Three days go by and Serena gets a manuscript in the post. It’s Tom’s story. He has attached a note: ‘Did I get this right?’ She reads it before going to work and is horrified to learn that alas, he did not get it right.”
I can see Victoria is getting a little bit restive. Must be her ADHD kicking in again, I thought.
“Do you want to walk around a bit and explore,” I ask Victoria.
“Sure,” she says. Victoria is always up for a little adventure.
“We’ll continue on with the story when we get back to the table.”
So, we pushed our chairs back and grabbed our drinks and went for a little trek about the place. We were sitting in the Library Room which was two stories tall and opened up to the ceiling. Up a flight of stairs there were two other rooms and a mezzanine looking over the downstairs portion of the library. Off to either side of the mezzanine were the two other two rooms, the Boudoir Room and the Fumoir Room, only there was no smoking in the Fumoir Room. What went on the Boudoir Room, I wasn’t certain. Each room was richly appointed with distinctive features of the period offering its occupants intimacy and privacy. Each had a maximum capacity of five. We took a peek in each room. Downstairs, off the bar, was a larger room called the Gaga Room which held up to 14 patrons. These rooms were offered for rent by the hour. In the bar area there were lounges made of richly upholstered plush red velvet with lamplight gently streaming over each one. One had the feeling that one could sink down into that velvet lining and disappear forever. We stood there transfixed for a while as if hypnotized by the ambiance. We snapped out of our reverie and headed back to the table in the library.
We sat back down and a little silence ensued as we thought about what we had just witnessed.
“Don’t you just hate that?” Victoria asked.
“That awkward silence when no one has anything to say?”
“Oh that. No, I don’t mind. Sometimes it’s good to just sit and think about things for a while and something naturally will come up of its own accord.”
“Well, I tell you what. Why don’t you tell me the rest of that story?”
“Good idea.” I raise my drink to her and say, “Here’s looking at you kid.”
She smiles back and touched her glass to mine as I resume the story.
“Now, where was I? Oh yes! Serena has just read Tom’s story and discovered to her horror that Tom indeed did not get the problem right. His story went something like this. A London architect suspects his wife of fooling around. One day, when he has time on his hands, he follows her to a sleazy hotel in Brighton. He spies her in the lobby with a man. They obtain a key from the desk clerk and head up the stairs. Terry, the architect, stealthily enters the hotel and follows them up the stairs, staying out of sight. They reach the fourth floor and Terry can hear a door open and close, but he can’t see which one. When he arrives on the floor, he can see there are only three rooms, 401, 402, and 403. His plan is to wait until the couple is in bed together then break into the room and catch them in flagrante delicto. Only one problem. Which room are they in?
“He listens for a sound but hears nothing. Time passes. He needs to make a choice. He chooses door 401 because it’s closest. He steps back to make a run for the door when the door to 403 opens and an Indian couple with a baby come out of the room. They smile at Terry and go down the steps.
“He figures he has a one-in-three chance his wife was in room 401. Which means that until now there was a two-in-three chance she’s in either 402 or 403. Now that he knows 403 is empty there must be a two in three chance, she’s in 402. Only a fool would stay with his first choice, for the laws of probability are immutable. He makes his run and crashes through the door of 402 and catches the couple in mid-stroke. He gives the chap a slap across the chops and make a hasty retreat out the door and heads for London to file for divorce.
“Serena thinks about this story all day long after she gets to work. It was a good story but it was flawed. It couldn’t stand as written. It didn’t make sense. The Indian couple coming out of room 403 did not tip the balance in favor of 402. Their emergence was random while Monty’s choice was not. He knows what is in each box. If Terry had chosen room 403 the Indian couple could not magically transfer themselves to another room so they could come out another door. After they come out of 403 Terry’s wife was just as likely to be in 402 or 401.
“Serena didn’t think she could just tell Tom the story didn’t work, rather she felt she had to fix it. She had an idea how. Tom could re-write the story and make it work. First, she had to get rid of the Indian couple. Then as Terry take a few steps back to run at the door to room 401, he overhears two housekeepers talking on the landing below. One says, ‘I’ll just pop upstairs and do one of the two empty rooms.’ The other says, ‘Be careful, that couple are in their usual room.’
“Terry quickly re-figures the odds and decides to stand in front of room 401 forcing the housekeeper to go into one of the other two rooms. She knows where the couple is. Whatever room she chooses, Terry will move to the other door, doubling his chances. And that is exactly what happens. The housekeeper goes into 403. Terry makes his move and crashes into 402 and voila, there they are!”
“And there you have it. The rest of the story!”
I finished my drink and the server came over and asked if I wanted another but I said no I’d had enough. Victoria declined as well. We spent the rest of the evening in pleasant conversation as is our wont to do and we were well positioned to engage in another one of our favorite pastimes, that of observing other customers and making fun of them or making up stories about their lives. We found this to be very amusing. Oh, I know, we were perfectly awful, but it was fun.
Later I got to thinking about that evening and thought it would be fun to reconstruct it as a story. I thought the parallels between the couples were interesting. Similar, but slightly different. Sort of like an alternate universe. There was magic in the invention. You take a little from here and a little from there and you take all the parts and put them together to form a comprehensive whole, synergized and harmonized. Sort of like a stew cooked by chefs to create something new and delicious. A story within a story, like the windmills of your mind. We had fun that night. And I vowed we would come back someday, no matter what it took. Come hell or high water.
7 thoughts on “The Monty Hall Problem”
I once explained the problem to a programmer friend. His solution to the counter-intuitive nature of the solution was to create a mini-program to mimic choosing thousands of times and then check the stats (an almost perfect 67%).
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Well, there you go!
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Oooo this is good! I like that kind of counterintuitive aspect, we did a little of that kind of thing in my psychology degree, the human mind trying to reconfigure a decision in light of new information but it’s pretty hardwired to certain lines of thought and we see what we expect so it takes a while to shift our processing. It sounded so ominous as you went down those stairs, I wasn’t sure whether you’d return! Love the names of the drinks, I could go for some Sparks Fly or Devil’s Advocate. On a side note, I’ve always wondered if people actually say “here’s to looking at you, kid”, or if it’s just in the movies. I’d like to say that to someone some day, or have someone say it to me. x
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Here’s looking at you kid….
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Thanks Caz! It’s always good to have an appeciative reader. I worked pretty hard on this was pretty happy with it. Here’s looking at you kid. X
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An interesting story! I would never have won any money on the Monty show!! The odds against me are way more than 50/50!!
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Haha! Thanks for your comments!
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