How do I calculate the odds of this highly, highly unlikely occurence?

Question

It was the birthday of a work colleague and so she was standing in line at a supermarket, buying a cake to share with the office. She happened to over hear the elderly lady ahead of her in the queue telling the clerk it was her 98th birthday that day. When she leant over to tell the lady they shared a birthday, the man in between them revealed it was his birthday too. They all showed each other I.D. to prove it then shook hands and decided to stay in contact.

What I want to know is, what are the odds of this happening to you on your birthday?

There seems to be three interlocking coincidences at work. Namely, that a random stranger in a queue will share your birthday, that a second random stranger in that queue will also share the date, and lastly, that you will all choose to tell each other.

Answers on a postcard please...

Answer 1

Well the first bit is fairly simple. There is a 1/365 chance that a random stranger will share your birthday (well actually 1/365.24 and a bit to allow for leap years). Given just three people standing in a queue the chance that they share the same birthday is therefore 1/365 x 1/365 = 1 in 133,225.

It gets more complicated if you allow for more people - if you've got 5 people in a queue, the chance that say 2 will share your birthday becomes much better.

As to the probability of them telling each other - that's anyone's guess.

Answer 2

Actually, there is a famous probability riddle called the "Birthday problem".  

In a group of 23 people, the chances of 2 people sharing a birthday is about 50%.  This seems high, but remember that you're not asking if 2 specific people share a birthday, but ANY two people in that group.  There are 253 possible pairings in a group of 23, which is more than half the number of days in the year.

Depending on how many people are in the queue, it's very possible that if those particular people in that queue didn't share a birthday, a different set of 2 or 3 people might have.

Considering that this was a line in a supermarket, where people probably buy birthday cakes every day, I'd suspect it happens more often than you think.  In most situations it may happen and we don't know about it, because the birth dates may not come up in conversation - but again, where people are buying cakes it may be a more commonly discussed topic.

Search at Wikipedia on "Birthday Problem" and you'll get a lot more statistics about this.

And why does my answer have to be on a postcard?

Answer 3

The odds are about 1 in 73. The situation comes about when 2 people share the same birthday so that is a forgone conclusion. The odds of 1 other person in the queue of say 5 people are 365/5 = 73. Not as uncommon as you think, especially as they were queing up fo birthday cake!

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