The cases of the Birthday Paradox and the Monty Hall problem.
Statistics, often regarded as a dry and seemingly unexciting field, has a knack for producing unexpected and fascinating results. It`s a discipline that bridges the gap between mathematics and the real world, offering insights and discoveries that can sometimes defy intuition. In this article, we will delve into the intriguing world of statistics, exploring how it can lead to surprising outcomes.
Imagine you are in a room with a group of people and you have this question:
How many of these people I need to have a 50% of probability that almost 2 of them shares the same birthday?
The answer may surprise you:
Just 23 people!
This is the heart of the mysterious “birthday paradox”
The birthday paradox is a fascinating example of how statistics can lead to seemingly unexpected results.
His solution defies intuition but is firmly rooted in probability theory.
To explain this phenomenon let’s start with a group of two individuals, the probability that the second person has the same birthday as the first person is 1/365, assuming a year has 365 days.
When we add the third person, we have to calculate the probability that he has the same birthday as at least one of the first two people.
Simply put, we can calculate the additional probability, which is the probability that both firsts two people do not have the same birthday as that third person.
The probability that the third person will “choose” a date different from the date of the first two people is 363/365 (because out of 365 there are 363 days left). So the probability that the first two people don’t have the same birthday as the third person is 363/365.
When we add people to the group, we need to calculate the probability that each new randomly selected person does not have the same birthday as someone who is already in the group in that moment.
When it comes to the 23rd person, the probability that none from the first 22 people will have the same birthday is 363/365 times 362/365 times 361/365, etcc… up to 344/365.
Therefore, the probability that none of the first 22 people will have the same birthday as the 23rd person is the product of these fractions.
Defining “n“as the number of people and “P “as the probability that none of the n person has the same birthday, we have:
Replacing n with 23, we have P(23) = 0.472.
The probability that at least two people have the same birthday is complementary to the probability above, which is equal to 1 minus the probability that none of the first 22 people has the same birthday as the 23rd person. This actually exceeded 50% when we reached 23 people. By calling S(n) the probability that at least two people have the same birthday:
S(n) = 1 — P(n)
S(23) = 1 — P(23) = 1–0.472 = 0.528 = 52.8 %
This discovery is unexpected due to the fact that we tend to think about “combination” rather than “comparison”. That is, instead of considering the probability of two group members having the same birthday, we focus on the number of possibilities of choosing pairs of people. This example teaches us the importance of understanding statistics and not making intuitive assumptions based on common sense alone. Probability can lead to surprising results, and the birthday paradox is just one example of how fascinating and mysterious the science of statistics can be.
The birthday paradox is attributed to Harold Davenport but the first publication about that is by Richard von Mises. The same Richard von Mises is the author of another example of how statistic can bring to an unexpected result: The Monty Hall Problem
The Monty Hall problem, a famous probability puzzle, is a difficult concept that has fascinated mathematicians, statisticians, and the general public. It is named after Monty Hall, former host of the American game show “Let’s Make a Deal”, where the issue was first raised. The puzzle became popular thanks to Marilyn vos Savant, who introduced it in her “Ask Marilyn” column in 1990.
The problem can be summarized as follows:
You are on a game show and facing three doors. Behind one of these doors is a valuable prize, such as a car, while the other two hide less desirable outcomes, usually goats. You make your first choice by choosing one of three doors. The host, Monty Hall, who knows what’s behind each door, then opens one of the two remaining doors, revealing a goat. At this point, you have a choice:
keep your original choice or move to another unopened door. The question is: what should you do to maximize your chances of winning the car??
This problem becomes interesting when we consider probability. Most people’s intuition suggests that there is a 50/50 chance that the car will park behind one of the remaining doors once the goat is revealed. However, the reality is completely different and counterintuitive.
The Monty Hall problem originates from a probability puzzle called the “three prisoners problem”, introduced in Martin Gardner’s “Mathematical Games” column in Scientific American in 1959. It was proposed by statistician Steve Selvin and published under the pseudonym “A.Selvin”. The three prisoners problem presents a similar scenario with three prisoners, one of whom is expected to be pardoned while the other two will be executed. The point is that prisoners have the right to guess their fate, and it turns out that changing your initial choice increases your chances of survival.
The version of the Monty Hall problem, now more widely known, was popularized by Marilyn Vos Savant’s column in Parade magazine in 1990. In his answer, Vos Savant gave Correctly argued that changing the gate would increase a competitor’s chances of winning the car at 2/3, while keeping the original choice will maintain a 1/3 chance.
The reason behind this counterintuitive result lies in conditional probability. When you make your initial choice, you have a 1/3 chance of choosing the car and a 2/3 chance of choosing the goat. When Monty reveals one of the goats, the information he provides is important. If you initially choose a car, there is a 100% chance that you will get a goat. However, if you initially choose a goat, there is a 100% chance that it will reveal the other goat, leaving the cart behind a closed door. Therefore, by switching, you are effectively taking advantage of the 2/3 probability associated with the original choice of a goat.
The Monty Hall problem continues to be a source of fascination in the fields of probability and statistics because of its ability to confound our intuition. It challenges our preconceived notions about chance and choice, illustrating how a deeper understanding of conditional probability can lead to surprising and unexpected outcomes. This is a classic example of how mathematics can provide information that seems counterintuitive but is true.
To put it briefly, the study of probability and statistics serves as a constant reminder of the ability of facts and mathematics to produce conclusions that go against common sense. Examples of situations that appear straightforward but can have unexpected and counterintuitive outcomes are the Birthday Paradox and the Monty Hall Problem. These puzzles serve as an example of how statistical thinking involves challenging our presumptions and revealing the intricate details of the world around us in addition to numbers and calculations. We can gain a greater knowledge of probability and the interesting but frequently perplexing field of statistics by accepting these unexpected results.
Bibliography
Siegmund D.O., Schreiber B., Augustyn A. Probability Theory, The birthday problem. (1999) Encyclopedia Britannica (https://www.britannica.com/science/probability-theory/The-birthday- problem#ref32767)
Probability and the Birthday Paradox (2012) Scientific American (https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/)
Pinker S. Why You Should Always Switch: The Monty Hall Problem (2021) Behavioral Scientist (https://behavioralscientist.org/steven-pinker-rationality-why-you-should-always-switch-the-monty-hall-problem-finally-explained/)
Mitzenmacher M, The Monty Hall Problem: A Study (1986) (https://web.mit.edu/rsi/www/2013/files/MiniSamples/MontyHall/montymain.pdf)
