Hairy People

Probability Level 1

Suppose that a human can have at most 1,000,000 hairs on his or her head.

In a city with at least 1,000,002 people, are there always at least two people with the exact same number of hairs on each of their heads?

Yes, always No, not necessarily

6 solutions

Denton Young
Jun 8, 2017

Relevant wiki: Pigeonhole Principle

There are 1000001 possible numbers of hairs a human can have on their head. (any number from 0 to 1000000 inclusive.) Since there are 1000002 people in the city, by Dirichlet's pigeonhole principle 2 of them must have a duplicate number of hairs from this set.

The question reads "ALWAYS TWO people". That is not necessarily true, as there could be MORE than two people with the exact number of hairs on their head. Poor wording. There are AT LEAST TWO people....

steven bengelsdorf - 3 years, 11 months ago

there could be a number of bald people with no heir

Vida Kudzma - 3 years, 11 months ago

ah, but if there were 2 people that were bald, they would both have 0 hairs, which means 2 people would have the same numbers of hairs

Jacob Huebner - 3 years, 11 months ago

what you are saying is correct. But we could have 1 million people with no hairs, and that is not 'exactly two'.

Raymond Li - 3 years, 11 months ago

@Raymond Li – Well, the question isn't asking for "exactly two" it's asking if there are 2 people; if there's more than 2 people there's still 2 people

Jacob Huebner - 3 years, 11 months ago

@Jacob Huebner – I think the question stating 'always 2 people' is causing the confusion

Raymond Li - 3 years, 11 months ago

@Raymond Li – That does appear to be the case, though it appears the question is intended to be read as "always at least 2 people"

Jacob Huebner - 3 years, 11 months ago

@Jacob Huebner – Fixed the wording.

Sharky Kesa - 3 years, 11 months ago

10% are bald men so you need another 100K hairy folk for truth to poevail.

Dale McAdam - 3 years, 11 months ago

The question is misleading as it implies that at least 1 hair is required to qualify, thus precluding those with no hair at all. So if there were three bald people they would not have the same number of hairs as they would have none!

Steve Pritchard - 3 years, 11 months ago

0 is a number and 0=0

Jacob Huebner - 3 years, 11 months ago

That was my logic also. Now if we were literally splitting hairs, one could argue that the word "number" can't refer to all numbers, because that could include non-integers, but rather to natural numbers. According to Wikipedia, there are differing opinions about whether or not 0 is a natural number - number used for counting.

Joni-Pekka Luomala - 3 years, 6 months ago

Joseph Gomes Short Films
Jun 18, 2017

If the first person has 0 hairs, the second has 1, the third 2, and so on all the way up to the millionth person we run out of options at 1,000,0002 people, because the most hairs a person can have is a million.

My issue with this is the fact it says "are there always two people...." Take the situation where everyone is bald, as a deliberately extreme case. In that instance, there are 1,000,002 people with the same number of hairs on their head. The question should say "are there always at least two people..."

Andy Sanderson - 3 years, 11 months ago

You misquoted the question, it says "are there always AT LEAST two people..." Makes a big difference, and solves your so called issue. You should know better as well Ray.

Joseph Gomes Short Films - 3 years, 11 months ago

Yes, but you didn't realise they changed the question after our comments were made. The question read 'always' two people beforehand.

So yes, I do know better....

@Jacob Huebner – Fixed the wording. – Sharky Kesa · 2 days, 19 hours ago reply

Raymond Li - 3 years, 11 months ago

@Raymond Li – Ok fair enough (:

Joseph Gomes Short Films - 3 years, 11 months ago

completely agree with Andy

Raymond Li - 3 years, 11 months ago

Abidur Rahman
Jun 8, 2017

I could explain it, but this video does a better job.

https://www.youtube.com/watch?v=HMpSKjK9clA

"How Many Humans Have the Same Number of Body Hairs? | Infinite Series | PBS Digital Studios"

Can you give us a small summary of what the video is about? That would be very nice

Agnishom Chattopadhyay - 3 years, 11 months ago

Similar to the birthday problem. Once you have 366 people, then there will be two with the same birthday.

Stephen Garramone - 3 years, 11 months ago

Hmm, that is true but the Birthday Problem is supposed to be illustrating how quickly the odds of having two people in the same day increase. However, the pigeonhole principle is a non-stochastic version of that.

Agnishom Chattopadhyay - 3 years, 11 months ago

You're neglecting the people born on February 29th!* I was born in a leap year so I've known two such people. For the pigeonhole principle to apply you need 367 people. *I'm assuming that's the error.

Richard Desper - 3 years, 11 months ago

Angel Krastev
Jun 21, 2017

Let's assume that each and everyone knows the number of hairs on his/her head. Let's assume we can write the numbers from 0 to 1000000 on the road. Let's assume that each and everyone can step on the number that matches the number of his/her hairs (nobody stays at home). There are two cases: 1. There will be some numbers without a person - which shows that there will be some numbers with more than one person; 2. The worst case scenario is when all numbers from 0 to 1000001 are occupied - that is the case when the person 1000002 with number of hairs from 0 to 1000000 must go to an occupied place. This is the Dirichlet's pigeonhole principle.

Phonex Ali
Jun 22, 2017

Pigeonhole Principle:
n people, k distribution n > k --> at least two people have same distribution

Ethan Song
Jun 20, 2017

Anyone could have between 0 and 1,000,000 hairs on their head;

Therefore there could be 1,000,001 people with different amounts of hair;

However, there are at least 1,000,002 people in the city, so there will always be at least two people with the exact same number of hairs on their heads, if not more.

Hairy People

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