Imagine that you are in a television contest.
You have a chance to win a brand new car, but to get it, you must chooser the door in which are the keys of in between three options what you have in front of you
Behind the other two doors is a prize that would comfort no one: a goat.
suppose that you choose door 2, the one in the middle, which gives you a hunch. So the presenter wants to help you and add excitement to the contest by revealing to you what’s behind one of the doors (he already knows where the goats are and which one is the car).
So open door 3, where a grumpy goat that has been locked up for a while appears.
The presenter makes you an offer: Do you keep the door 2 that you chose? Or do you prefer to change it to door 1?
If you think about it a bit, there are two possibilities left and you might assume that there are 50% of odds that you hit and win the car, and 50% that you go home with a goat.
But if you think about itas it did Marilyn vos Savantyou would have almost a two-thirds chance (66%) of winning the car.
And it all has to do with something as simple or as complicated as a probability analysis.
Guinness of intelligence
Marilyn Mach vos Savant, now 76, was a longtime columnist for “Ask Marilyn” (Ask Marilyn), a space in the US press where he answered questions, riddles and offered his points of view on many topics.
She is also the author of fiction and nonfiction books, and was an investment entrepreneur. But since she was a child she had another more notable title.
After doing a couple of tests intelligence quotient (IQ)in one of them he obtained a score of 228more than double the average.
the book of Guinness World Records recorded her from 1985 to 1989 as the woman with the highest IQ on record. That’s why she was called “the smartest person in the world“.
Vos Savant is the daughter of European immigrants who settled in the city of St. Louis, Missouri, in the central US, where she was born in August 1946.
She was convinced that people should have both parents’ last names. And for that reason she adopted the maiden name of her mother, Savant, which curiously means “wise person” in French.
As he progressed in his education, he excelled in math and science at his school. On her 10th birthday, she took the Stanford-Binet and Hoeflin’s Mega IQ tests. It was in the latter that she obtained the score of 228 that the Guinness organization took into consideration.
Since then she was considered a child prodigy, although that did not change her lifestyle much. In her adolescence, she has recounted, she helped in her parents’ store and she liked to read a lot.
And far from being interested in some prestigious university in the Ivy League de EE.UU., he chose to study philosophy at Washington University in his native St. Louis. However, he abandoned his studies to engage in a family investment business.
But her fame for the intelligence test followed her.
In the 1970s, she generated enough money to self-finance her desire to be a writer. She participated in published intelligence tests, such as the Omni IQ Quiz Contest, as well as her own literary works and articles in magazines and newspapers.
After moving to New York, he appeared on David Letterman’s all-star talk show and Joe Franklyn’s. He almost always received questions about what intelligence is.
“Intelligence would be your overall ability to take advantage of experience”, exposed Franklyn. “IQ could, at most, measure your ability to use that intelligence.”
At that time he created the “Ask Marilyn” column, distributed nationally in the magazine Parade which was included in many newspapers.
It was in 1989 when he was asked about the dilemma of which door to choose to win the prize. Her response caused a stir, even among statisticians and scientists.
The answer to the riddle
The idea on which the riddle of the car and the goats is based was nothing new when it came to Vos Savant’s column.
Since a couple of decades earlier, he was known as “The Monty Hall Problem” by the name of the host of the American TV show Let’s Make a Deal (“Let’s make a deal”) where similar situations arose.
Statistician Steve Selvin, among others, presented a solution in the academic journal American Statistician in 1975.
But it was Vos Savant’s response, very similar in logic to Selvin’s, that caused the stir.
«Yeah; you should change. The first gate has a one-third chance of winning, but the second gate has a two-thirds chance. This is a good way to visualize what happened. Suppose there are a million doors and you choose door number 1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens all of them except door #777,777. You change the door quickly, don’t you? », Said his answer.
His statement produced a shower of responses. Vos Savant said that received about 10,000 letterss, including some 1,000 mathematicians and doctors in various disciplines, reviewed The New York Times in a 1991 article.
“You messed it up!” said Robert Sachs, a professor at George Mason University, Virginia. “As a professional mathematician, I am very concerned about the general public’s lack of math skills. Please help by confessing your mistake and in future be more careful«.
For a time, Vos Savant defended his answer, despite criticism.
“You are dead wrong,” wrote E. Ray Bobo, a professor of mathematics at Georgetown University. “How many angry mathematicians does it take to change your mind?”
I was right?
Vos Savant’s answer is correctas long as be fulfilled for the presenter to reveal what is behind the wrong door and offer the opportunity to change. That is why this problem belongs to the branch of the conditional probability.
By making the door pick, you start the contest with a 1/3 chance of winning. The other 2/3 are under the control of the presenter. You may have chosen the right one, but you still only have a 33% chance of success.
When the presenter reveals one of the wrong options, if you change your choice you add another third of possibilities in your favor (66%).
The usual mistake is to assume that you have a 50% chance at that moment, since a condition has already been met (the opening of one of the doors) that generates a new scenario.
The fact of changing the door It does not guarantee that you will win the car, it only increases the chances as long as the variable that the presenter opens a door is met, which will never be the one with the car keys, but rather a goat.
This has been proven multiple times. A few years ago, the BBC was part of the experiment in which Cardiff University students were divided into presenters of the show and contestants.
Those who switched were roughly twice as successful, as among 30 contestants who decided do it18 won the car. That is, there was a 60% hit rate. Meanwhile, of 30 who decided to keep their choice, there were only 11 correct answers, a rate of 36%.
Vos Savant did not receive as many letters of apology as criticism, but one did, from Professor Sachs who said: ‘Now I am eating the humble pie. I promised as penance to respond to all the people who wrote to reprimand me. This has been a deep professional shame ».
Others argued that the initial question from the reader who wrote to “Ask Marilyn” never specified that the host should necessarily offer a change or reveal a door.
In Let’s Make a Deal, monty hall indeed he had at his discretion to offer the change o noand even went so far as to add one more variable: cash to tempt the contestant to take their first choice, and the more it offered, the more it motivated the contestant to switch doors and fail.
Giving everyone expensive gifts, like a car, was not the intention of the show. And Hall was the one in charge of the situation.