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Birthday paradox

By David, 3 September 2013 Activity

How likely is it that at least two people out of 30 will share a birthday?

Three pieces of paper. Jane Doe 29 February, Timothy Tam 18 April, John Smith 22 April.

Collect each person’s birthday on a separate piece of paper and put them in order.

You will need

  • Around 30 people such as a class or your extended family. You could have more or fewer than that and they don’t all need to be in the same place!
  • Paper and pens for each person

What to do

  1. Tell each person in your group that you are conducting a survey and that you’d like them to give you some information.
    If you collect around thirty birthdays, you've got a good chance of finding two on the same day.

    If you collect around thirty birthdays, you’ve got a good chance of finding two on the same day.

  2. Ask them to write their name and birthday on a piece of paper and then collect all the papers. Don’t forget to thank them for their help!
  3. When you have a piece of paper from each person you are surveying, sort them into piles based on the month of their birthday. Put everyone with a birthday in January in one pile, February birthdays in another, March birthdays in a third, and so on.
  4. Now sort each pile so that the birthdays are all in order from earliest to latest, so a birthday on 2 January comes before 16 January.
  5. On a sheet of paper, write everyone’s names and birthdays in order from 1 January through to the 31 December. You can use this list so you never forget a birthday again!
  6. Do at least two people in your group share a birthday?

What’s happening?

If you surveyed 30 people, you are quite likely to have found that at least 2 shared a birthday. This is probably more likely than you think.

A cupcake with a candle on it sitting on top of the paper of names and dates.

With your new list, you’ll never miss someone’s birthday ever again!

There are 366 possible days to have a birthday, which means the chance of any person having their birthday on any particular day is around 1/366. If you’ve only surveyed 30 people, it might seem like you’ve got a 30/366 chance of getting two people having the same birthday, which would be less than a 10% chance. However, instead of looking at individual birthdays, you should be looking at pairs of birthdays.

Each person in your group could share their birthday with any other person in the group. With 30 people there are 435 possible pairs of people, and each pair has around 1/366 chance that they are both on the same day. This adds up to chance of around 70% that there will be two people sharing a birthday. You only need 23 people to get a fifty percent chance that there are two people sharing a birthday.

Applications

The birthday paradox shows that random events often bunch up unexpectedly. This is very useful if you need to work out if a list of numbers has been made up.  It is very hard to make up a set of random numbers – that is numbers that have no overall pattern. This fact can be used to detect possible fraud.

In many sets of numbers, including stock prices and electricity bills, the first digit of a number is most likely to be 1. In fact this happens about 30% of the time. This rule was discovered by Simon Newcomb in 1881, and rediscovered by Frank Benford in 1938.

If a set of numbers has been made up, it is not likely to follow this rule. Investigators look at patterns of numbers to detect evidence of fraud that may require further investigation.

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5 comments

  1. Sorry, but could you explain how you get to “around 70%”?

      Reply
    1. The way described above isn’t the best way to attack the problem mathematically. However, it’s a good way to see the difference between the intuitive answer and the mathematical one.

      The following calculation is not rigorous, but you can still get a good estimate using this attack – we’ll calculate the chance that all the birthdays are different.

      Each pair has a 365/366 chance of being different, there are 435 pairs, and all pairs need to be different. so the probability that all the pairs are different is:

      (365/366)^435 ~ 0.3

      So a 30% chance they’re all different, 70% chance there’s at least two people who share a birthday.

      As I said, this method isn’t actually correct, but it’s pretty close. A better analysis is available on Wikipedia:

      http://en.wikipedia.org/wiki/Birthday_problem

        Reply
    2. because if you have 23/366=50% so 30 is 23 + 7=30 so that =70%

        Reply
  2. Thanks, quite fascinating. I knew it wasn’t simple.

      Reply
  3. There is one thing i knew about this…

    it IS a paradox…

      Reply

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