Math Series Part VI: The Birthday Paradox
The Birthday Paradox
People become astounded when they encounter something that is totally against what they perceive to be normal; when they encounter something that is against their intuition. Aspects of mathematics and probability tend to do this to people. It is the same reason why so many people play the lottery when the actual chance of selecting the right combination of numbers is one from mullions of combinations. We want to believe that we have special attributes and that our individuality is unique and incomparable—and that we can be the one in the millions who can win the lottery.
It is this form of vanity and tunnel vision that brings about the birthday paradox. It states that in a room filled with less than 365 people, there is still a very a high chance that someone shares the same birthday (day and month) with someone else. Now, with 350 people in the room, this may seem very likely. But, if we were told that in a room with just yourself and 22 other people that there was almost a 50-50 chance of two of us sharing the same birthday, the notion seems a little hard to believe. Yet, this is where mathematics takes over to turn the issue from one of common sense to a more numerical and mathematical approach, and where the so-called ‘paradox’ ceases to exist.
Indeed, when we consider the probability of you having your birthday, it is 365/365, as you can be born on any day of the year (and here we are ignoring leap days for simplicity). However, the probability that someone else shares this day with you is 1/365 = 0.27397…%. So, in a room with 22 others, you have 22 people to compare with, so this probability increases to 22/365 = 6.02739…%. That’s clear enough. However, what we are really considering is not just us ourself. We are thinking about the chances of two people from the entire room sharing the same birthday between them. This changes the odds that we are calculating.
Indeed, if we transform the calculation we are making to think about the chances of somebody not having the same birthday with someone else, the odds are 364/365. What we have to consider now is the chances of everyone in the room sharing a birthday with someone else. We know that the first person will have 22 people to compare to, but the second will now only have 21 because they have already been compared to the first person. This is true for the third person too, as they only have 20 new people to compare to. This goes all the way down to the last person, who will have only 1 person to compare to. Thus, in the entire room of 23 people, we can generate 22 + 21 + 20 + 19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 253 comparisons. Hence, if we can effectively compare birthdays 253 times, we can calculate the probability that no one will share a birthday, and this is the 364/365 odd multiplied by itself 253 times; i.e. (364^253)/365 = 0.49952... This means that in a room of 23 people, there is a 49.95% chance of none of them sharing a birthday. And this also means that there is more than a 50% chance of two of them sharing a birthday.
- Mathematical Wonders Series: Part I
In Part I of this series, we look at the fascinating Möbius band
- Mathematical Wonders Series: Part II
In Part II of this series, we look at the incredible act of doubling.
- Mathematical Wonders Series: Part III
In Part III of this series, we look at the use of Data Mining.
- Mathematical Wonders Series: Part IV
In Part IV of this series, we look at the magic of the number phi.
- Mathematical Wonders Series: Part V
In Part V of this series, we look at alternative ways to divide big numbers.
Removing The Self
The probability approach and the removal of the individual from this calculation helps to clarify how there is really no paradox at all. There is merely the need to remove the ‘self’ from the calculation and to observe how each individual in the group relates to every other individual in the group, which gives us this 50-50 chance when there are 23 people. Indeed, these odds would change given the number of people in the room. In fact, with just 70 people in the room, the odds of two people sharing a birthday are 1 - (364^2485)/365 = 99.89056...%, as there are 2485 comparisons to be made between the 70 people. It is almost certain, then, that two people in a room of 70 will share a birthday, and then the odds only get better from there.