### The Birthday Paradox

I bet you are not going to believe the following statements.

*There is a 50% chance for at least two persons in a group of 23 persons at a party to have the same birthday! *

* **When there is a group of 50 people, such chance is even very much higher, at 97%!*

Is this true? Do you want to check and confirm the trueness of these statements during your next partying?

Let’s reason out this phenomenon through the basic rules of probability.

Remember that if you flip a perfect coin, the probability of getting a head is 1/2 since one out of two possible outcomes is a head. If you flip a coin twice, the probability of getting both heads is (1/2)(1/2) or 1/4 or 0.25.

On the other hand, if a million people buy a ticket in the lottery draw and you buy one, then, the probability of you winning the first prize is 1/10^{6} or 0.000001 or 0.0001%. That chance sounds miserable, of course.

Now, there is a rule of probability known as the complement rule, which says the probability of something happening is one minus the probability it does not happen. So, the complement rule says the probability of not winning the lotteries is (1 – 0.000001) or 0.999999. That means the chance of winning is very slim indeed.

Let’s assume that we have 365 days in a year and so the probability that two persons have different birthdays is (365/365)(364/365) or 0.997. This calculation is understandable: the first person can have his or her birthday any day of the year and so the first factor is 365/365 or 1, whereas the second person must have his or her birthday on one of the remaining 364 days out of 365 days in order to be distinct from the first guy, giving rise to the second factor 364/365.

Hence, we can conclude that the probability that both persons have matching birthdays on any of the days in a calendar year is (1 – 0.997) or 0.003 or 0.3% chance. When we have 3 persons for consideration, the probability of having distinct birthdays is (365/365)(364/365)(363/365) or 0.992. As the number of people increases, the probability of having different birthdays decreases as shown in the table below:

No. of persons | Probability of no matching birthdays | Probability of at least one matching birthday |

2 | 0.997 | 0.003 |

3 | 0.992 | 0.008 |

4 | 0.984 | 0.016 |

5 | 0.973 | 0.027 |

6 | 0.960 | 0.040 |

7 | 0.944 | 0.056 |

8 | 0.926 | 0.074 |

9 | 0.905 | 0.095 |

10 | 0.883 | 0.117 |

11 | 0.86 | 0.14 |

12 | 0.83 | 0.17 |

13 | 0.81 | 0.19 |

14 | 0.78 | 0.22 |

15 | 0.75 | 0.25 |

16 | 0.72 | 0.28 |

17 | 0.68 | 0.32 |

18 | 0.65 | 0.35 |

19 | 0.62 | 0.38 |

20 | 0.59 | 0.41 |

21 | 0.56 | 0.44 |

22 | 0.52 | 0.48 |

23 | 0.49 | 0.51 |

30 | 0.29 | 0.71 |

35 | 0.19 | 0.81 |

50 | 0.03 | 0.97 |

100 | 0.0000004 | 0.9999996 |

From the above table, we have shown that in a group of 23 people, the probability of matching birthdays is 0.51 or 51%, which is almost the same as flipping a coin! And, with 50 people, there is a 97% chance at least two persons have the same birthday and for 100 people, it is almost 100% sure of birthday matching!

In fact, by this reasoning, we see that when there are 366 people or more under this consideration for 365 days (no leap year, please), certainly the probability is perfectly 1 or 100% chance. It is just like putting 366 pigeons in 365 holes. There are bound to have one box with more than one pigeon inside. This is called the *Pigeonhole principle*.

The above illustration is called the Birthday Paradox. Why do we call it a paradox?

Wikipedia says that a **paradox** is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time. In other words, a paradox refers to trying to solve a problem when certain aspects of the problems are overlooked.

The reason that the probability of matching birthday is greater than what our intuition tells us is due to the fact that we have overlooked all the scenario that birthdays can match. When we say “matching birthdays” here, we only mean matching birthdays of *any two* persons, *any day* of the year, and nothing more. Definitely, there are more of these probabilities than what we have considered.

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