## Same Birthday

Posted by Karl Sharman on July 10, 2010 – 1:10 pm

How many people would you need to ask at random before there is a-better-than- average chance that at least two of them have the same birthday?

How many Martians would you need to ask at random before there is a-better-than-average chance that at least two of them have the same birthday? I’m going to offer a little assistance on this part of the puzzle:

1. Martians do exist

2. The Martian year is 687 days long, nearly.

July 10th, 2010 at 3:40 pm

23 Earthlings, 32 Mars bars

July 10th, 2010 at 9:02 pm

How many days in a year to need 42 people?

July 11th, 2010 at 12:31 am

really??

i had 30+ people in my class in grades 1 thru 8,different people every time,no one shared my b-day!

explain.

July 11th, 2010 at 1:25 am

Chris, To help add to your collection of problems for which the answer is 42, I’d say 1200 days in a year.

That’s a long time to wait for your birthday to come round.

July 11th, 2010 at 2:34 am

Hi Knightmare. That’s a different thing. The odds would have been better than 70% that in each class (separately) that two people had the same birthday. But the odds would only be around 8% that someone had the same birthday as yours – I’m assuming that yours isn’t the 29th of April.

Do a Google with “shared birthday probability”.

Thanks Wiz. I just found and used an approximation formula that gave 1242.16

July 11th, 2010 at 3:23 am

Wiz. On planet 42, the day is only one Earth hour long.

July 11th, 2010 at 7:30 am

There was some maths and a little reasoning I was expecting to see to quantify the answers…… So, for the sake of completeness, here’s how we do it.

With one human there is a 0 percent chance that you’ll have two humans with the same birthday.

With two humans the probability that they won’t share a birthday is 364/365. The probability that they will share a birthday is therefore 1 – (364/365).

With three humans the probability that they won’t share a birthday is the same as for two humans, times 363/365. So the probability that three humans will share a birthday is 1 – (364/365) * (363/365). Notice that with each additional person added, the probability that he or she shares a birthday with one of the previous persons goes up, because there are fewer “free” days remaining.

Following this progression, the probabilities are:

Number of Humans Probability of two shared birthdays

1 0

2 0.00273972602739725

3 0.00820416588478134

4 0.0163559124665502

5 0.0271355736997935

6 0.0404624836491114

7 0.0562357030959754

8 0.074335292351669

9 0.0946238338891667

10 0.116948177711078

11 0.141141378321733

12 0.167024788838064

13 0.194410275232429

14 0.223102512004973

15 0.252901319763686

16 0.28360400525285

17 0.315007665296561

18 0.34691141787179

19 0.379118526031537

20 0.41143838358058

21 0.443688335165206

22 0.47569530766255

23 0.507297234323986

So 23 humans will have a better than average chance of sharing a birthday.

The Martians, who have a year of nearly double ours? Surprisingly, only nine more Martians will be needed.

And Chris…. it’s “unneeded”;-)

July 11th, 2010 at 10:01 am

Ooops, unneeded it is. Now I knows.

July 11th, 2010 at 11:21 am

Karl, the Martian year is 668.6 Martian days. Taking that to be 668 days, we need 31 Martian peeps needed – not the 32 that 687 days/year requires.

On planet 42, there are 1242.16 days/year.

July 11th, 2010 at 11:31 am

Chris, my info, link attached, quotes 687 days, although for 668 days, you are correct.

http://www.universetoday.com/guide-to-space/mars/how-long-is-a-year-on-mars/

FIrst para: “A year on Mars lasts 687 Earth days. In other words, a Martian year is almost double an Earth year. When you celebrate your 25th birthday on Mars, you’d actually be 47 years old if you’d been living on Earth….”

However, Wiki.Answers gives this response:

http://wiki.answers.com/Q/How_long_is_a_Martian_year

A year on Mars is almost twice as long as a year on Earth, being 686.9726 days long. However, this would be in 24 hour “Earth days”, forty minutes longer than the Martian day which is actually called a “sol”. There are 668.5921 sols in a Martian year.

Hmmmmmmm

July 11th, 2010 at 8:02 pm

Aaaaaaaaaaaaah! Brain explosion!

July 14th, 2010 at 11:02 am

Terrific work! This is the type of information that should be shared around the web. Shame on the search engines for not positioning this post higher!