How was this probability calculated?

[plindboe]

I have always been great at math, but unfortunately I'm terrible at statistics, and wish to learn more. This one made me curious as to how it was calculated:

It can be calculated that if you take just 23 people, the chance that two have their birthday on the same day is 50%. Yet, people at a party who meet someone that shares their birthday will often think this is uncanny.

From here: http://www.brainyencyclopedia.com/encyclopedia/m/ma/magical_thinking.html

(A nice little article btw. Take a look)

 

[The Don]

Two people, chance that they don't share a birthday is 364/365

Three people, they all need to have unique birthdays, chance of this is 364/365 * 363/365

Four people they all need to have unique birthdays, chance of this is 364/365 * 363/365 * 362/365

n people they all need to have unique birthdays, chance of this is 364/365 * 363/365 * 362/365......*(366-n)/365

The point at which this geometric series is less than 0.5 is the point at which the chances of any two sharing a birthday is more than 50%.

 

[epepke]

Two people, chance that they don't share a birthday is 364/365

Three people, they all need to have unique birthdays, chance of this is 364/365 * 363/365

Four people they all need to have unique birthdays, chance of this is 364/365 * 363/365 * 362/365

n people they all need to have unique birthdays, chance of this is 364/365 * 363/365 * 362/365......*(366-n)/365

The point at which this geometric series is less than 0.5 is the point at which the chances of any two sharing a birthday is more than 50%.

Yeah.

This is a general technique that can make nearly all probability problems easier. If it isn't obvious how to calculate the probability that it's true, calculate the probability that it's false and subtract from 1.

 

[plindboe]

Great thanks Don.

 

[CurtC]

Yet, people at a party who meet someone that shares their birthday will often think this is uncanny.

I have a hard time believing that people will think this situation is "uncanny." If you attend a party with 20 other people, there's about a 5% chance that someone there will share your birthday. Would someone really be shocked, shocked, to find that a 5% probability had occurred?

 

[drkitten]

Re: Re: How was this probability calculated?

I have a hard time believing that people will think this situation is "uncanny." If you attend a party with 20 other people, there's about a 5% chance that someone there will share your birthday. Would someone really be shocked, shocked, to find that a 5% probability had occurred?

I might be if I had misjudged the odds. What's "uncanny" is not that a 5% probability occurs, but that the odds are unintuitively high.

 

[Iconoclast]

Re: Re: How was this probability calculated?

If you attend a party with 20 other people, there's about a 5% chance that someone there will share your birthday. Would someone really be shocked, shocked, to find that a 5% probability had occurred?

Well that's a different probability problem altogether.

To solve that one, first find the probability that not one of the people at the party plays the lottery every week, then...

 

[T'ai Chi]

I have always been great at math, but unfortunately I'm terrible at statistics, and wish to learn more. This one made me curious as to how it was calculated:

Check out this applet for some fun. Try running it with some number of birthdays, say 50 or above. You'll almost always get 1 or more occurances of 2 people sharing the same birthday.

 

[Capsid]

I was in a chat room the other night with 3 other people. I shared my birthday with one and the other two shared theirs.