This post has the purpose of presenting a result that may seem counterintuitive and that can provide a really nice excuse for a wager between you and one or more of your friends. For this post, when I talk about a birthdate I am only referring to the day and month of birth, and not the year.
What is the probability that you and your best friend have the same birth day and month? Even without an exact number one knows that you are much more likely to have different birthdates than having equal birthdates. Assuming all 366 days are equally likely, the probability that two people have the same birthdate is 1366≈0.27% and the probability that the birthdate is different is 365366≈99.73%. How many people do you need so that the probability of existing at least two sharing the birthdate is higher than the probability of everyone having different birthdates?
What would your guess be?
It only takes 23 people. If you have a group of 23 people, the probability that no one shares the birthdate is approximately 49.37%! That is the same as saying that, in a group of 23 people, there is a ≈50.63% chance that two people share a birthdate.
This seems very counterintuitive because 23 people can only cover 23 of the 366 possible days, which represents a ratio of 23366≈6.3%, a very low number which wouldn't make us think that 23 people were enough to make this happen. But why would you care?
Whenever you find yourself in a group of 23 people or more, you can bet the other people of the group that at least two of you share the same birthdate. If you do this often enough, you will make money! Just like a casino: you will win some and lose some, but in the long run you are expected to profit from this.
How can you compute these probabilities? Take the n people of your group and line them up. We will be comparing the birthdate of a person with the birthdates of everyone to the left. What is the probability that the second person has a birthdate distinct from the first person? It is 365366. What is the probability that the third person has a birthdate distinct from the two people to the left, if the two to the left have distinct birthdates? It is 364366, so what is the probability that the first three people have distinct birthdates? It is 365366×364366, the probability that the second person doesn't share its birthdate with the first person times the probability that the third person doesn't share its birthdate with the first or second people.
We can keep this train of thought going: if the first three people have different birthdates, what is the probability that the fourth person has a distinct birthdate? It is 363366, so the probability that the first four people have different birthdates is 365366×364366×363366.
In a general setting, the probability that n<366 people have distinct birthdates is n−1∏i=1366−i366
For your convenience, I included a window with that formula implemented, you just have to write the number n where the 23 is and then a number will be printed to the console: the probability that, in a group of n people, all of them have distinct birthdates.
Leave your feedback in the comments section. Did you know this already?
What is the probability that you and your best friend have the same birth day and month? Even without an exact number one knows that you are much more likely to have different birthdates than having equal birthdates. Assuming all 366 days are equally likely, the probability that two people have the same birthdate is 1366≈0.27% and the probability that the birthdate is different is 365366≈99.73%. How many people do you need so that the probability of existing at least two sharing the birthdate is higher than the probability of everyone having different birthdates?
What would your guess be?
It only takes 23 people. If you have a group of 23 people, the probability that no one shares the birthdate is approximately 49.37%! That is the same as saying that, in a group of 23 people, there is a ≈50.63% chance that two people share a birthdate.
This seems very counterintuitive because 23 people can only cover 23 of the 366 possible days, which represents a ratio of 23366≈6.3%, a very low number which wouldn't make us think that 23 people were enough to make this happen. But why would you care?
Whenever you find yourself in a group of 23 people or more, you can bet the other people of the group that at least two of you share the same birthdate. If you do this often enough, you will make money! Just like a casino: you will win some and lose some, but in the long run you are expected to profit from this.
- In a group of 23, your chances of winning are above 50%;
- In a group of 27, your chances of winning are above 60%;
- In a group of 30, your chances of winning are above 70%;
- In a group of 35, your chances of winning are above 80%;
- In a group of 41, your chances of winning are above 90%.
How can you compute these probabilities? Take the n people of your group and line them up. We will be comparing the birthdate of a person with the birthdates of everyone to the left. What is the probability that the second person has a birthdate distinct from the first person? It is 365366. What is the probability that the third person has a birthdate distinct from the two people to the left, if the two to the left have distinct birthdates? It is 364366, so what is the probability that the first three people have distinct birthdates? It is 365366×364366, the probability that the second person doesn't share its birthdate with the first person times the probability that the third person doesn't share its birthdate with the first or second people.
We can keep this train of thought going: if the first three people have different birthdates, what is the probability that the fourth person has a distinct birthdate? It is 363366, so the probability that the first four people have different birthdates is 365366×364366×363366.
In a general setting, the probability that n<366 people have distinct birthdates is n−1∏i=1366−i366
Recall that we have assumed that a person is equally likely to be born in any of the 366 existing days, which isn't exactly right (if I had to guess, the 29th of February would be one of the less likely days for example). However, assuming equally likely birthdates gives the worst case scenario for you, the one betting that there are people sharing a birthdate.
For your convenience, I included a window with that formula implemented, you just have to write the number n where the 23 is and then a number will be printed to the console: the probability that, in a group of n people, all of them have distinct birthdates.
Leave your feedback in the comments section. Did you know this already?
- RGS
Pocket maths: the birthday bet
Reviewed by Unknown
on
June 14, 2018
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