All animals are equal, but some animals are more equal than others

Probability Level 2

There are $51$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?

Details and Assumptions

Assume the probability of each person's birthday is random and independent from one another.

The answer is 5.

5 solutions

Caleb Townsend
Feb 22, 2015

The least number of birthdays in the same month is obtained when the birthdays are evenly distributed by month; that is, there are roughly the same number of birthdays each month. We can determine this smallest number using the Pigeonhole principle .

Note that $51 = 4\times 12 + 3,$ so at bare minimum, there will always be $4$ people with birthdays in the same month. There are still $3$ people left over, though, and their birthdays must fall in some month of the year. So in reality, there must be at least $\lfloor \frac{51}{12} \rfloor + 1$ birthdays in at least $1$ month.

Therefore the answer is $n = 4 + 1 = \boxed{5}$

Almost. You need to show that $n \ge 6$ is impossible, but this is easy; just let five people each having birthdays in the first three months and four each for the rest.

Ivan Koswara - 6 years, 3 months ago

I'm not agreeing with this. I'll explain why. I think the answer is 3, for this reason. There are 52 weeks in the year and we have 51 people. So we have 51 weeks, with each person born in a different week and one empty week (with no one born in that week). Those weeks are then divided into 12 months. At least one of those months will have 4 weeks (say February) and if the empty week is also in that month, then instead of 4 people born in that month, there will 3 people born in that month. (There are actually 53 full or partial weeks in a year, but I don't think the partial week (1 or 2 days) affects the solution.)

Jerry Kitich - 5 years, 5 months ago

@Jerry Kitich , the problem with your solution is you incorrectly assumed all birthdays are in different weeks. Nowhere in the question does it say this. There is nothing stopping the birthdays from being in the same week of the month, or the same day for that matter.

Peter Silke - 4 years, 8 months ago

But i think that 4 must be the correct answer provided that there are at least 4 births in every month, and the statement says for which value of n is this always true. and 5 is not true for every month. If I am wrong, please forgive my lack of logic, and explain it for me, thanks beforehand.

Reynaldo C. Del Castillo - 5 years, 5 months ago

completely agree

Joshua Coddington - 5 years, 5 months ago

The stipulation that the birthdays are random and independent is unnecessary.

Anthony Cutler - 3 years, 7 months ago

Why did you take floor of 51/12?

Anoubhav Agarwaal - 2 years, 9 months ago

You might find this helpful.

Prasun Biswas - 1 year, 4 months ago

Correlating to the pigeonhole principle, we have the people as pigeons and the months as pigeonholes.

Prasun Biswas - 6 years, 3 months ago

Using the picture as a hint eh.

Shilong Zhu - 6 years, 3 months ago

Otavio Monteagudo
Feb 27, 2015

12 months, 51 people. 12 x 4 = 48, so if birthdays are equally distributed, 4 people will have a birthday on each month. Since 51 > 48, each month will have at least 4 people with a birthday on it, but there will necessarily be a month with more than 4 people born on it. Since we need the largest value for 'n' and 'n' must be always true, let's assign a new person to each month; not every month will have 5 people having birthdays on it, but 3 other months will.

Bang Quang
Mar 4, 2015

Simple that we should find maximum value in the minimum case: To get least people in same month, we should distribute equally, so n = 51/12 = 4 + 3/12, remain 3 peoples to distribute -> so in this case, we will get max 5 people in same month.

Hassan Hdach
Mar 6, 2016

See also https://en.wikipedia.org/wiki/Pigeonhole_principle?wprov=sfla1

Achille 'Gilles'
Dec 3, 2015

There will always be one month of the years with at least 5 anniversaries.

An almost even distribution would place 4 birthdays on 9 different months and 5 for the 3 others months.

Improbable but not impossible, all are born in the same month so the max would be 51.

All animals are equal, but some animals are more equal than others

The answer is 5.

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