# Safety in Numbers: The Birthday Paradox

This time, Safety in Numbers celebrates its birthday the only way it knows how – but it’s not the only one…

Numbers are typically pretty safe things. People know that big numbers can be good (credit ratings, bank balances or medals) or bad (cholesterol, criminal convictions or world rankings) but they usually give a fairly accurate representation of where you deserve to be.

Likewise, if you’re unsure as to whether to watch a film or not you’ll probably bow to Rotten Tomatoes or IMDB’s user and critic ratings to help inform your final decision. They may not always be right for you, but they give you an idea of what to expect.

However, there are occasions when numbers throw us an absolute curve ball. Even the best algorithms and experts in the world sometimes fail to collect or correctly interpret enough significant data to correctly predict a major event, such as Brexit or Donald Trump’s election win, resulting in the numbers being at odds with reality.

And another is with the Birthday Paradox.

The Birthday Paradox is more than slightly counterintuitive in that it goes against everything we assume about probability.

When you think of your birthday, there is a one in 365 (or one in 365.25 if you include leap years) chance of being born on any given day (barring seasonal variations such as Christmas and Valentine’s Day affecting baby-making activity). Therefore, if you were to meet a random stranger, they would have a one in 365 chance of being born on the same day as you. This means that there is only a 0.27% chance of the two of you sharing the same birthday. Not the most inspiring odds you could think of.

However, the more people you bring into the equation, the more chance you have of two people in the group sharing a birthday; add another eight people to the mix and the odds fall from one in 365 to one in nine (45 potential pairs out of 365).

This is where the “paradox” of this equation comes into play; when you tell someone that there is a 50% chance of two in 23 people sharing a birthday, they naturally assume that the birthday is theirs.

However, in this hypothetical room full of people there are actually a total of 253 potential pairs of dates, with every individual having 22 potential matches of their own. This takes the odds to a staggering 50% that two of these individuals will share the same birthday.

In a sample of 26 people – the number of people who sit with me in my corner of the office – the chance of sharing a birthday rises to 59%, which sounds more promising if you were to put money on it.

However, today we will be celebrating not two but three birthdays. While you only need a sample of 23 to gain a 50% probability of two of them sharing a birthday, this sample required for there to be a 50/50 chance of three of them to be born on the same day leaps to 87, meaning that it is fairly staggering that 11.5% of my colleagues to have their birthdays on 20th October.

This is made more remarkable by the fact that 20th October isn’t even one of the most common dates to be born on, according to the ONS, with an average of 1,790 daily births between 1995 and 2014 putting it as the 236th most common day to be born. Essentially, this means that 20th October only narrowly escapes the bottom tertile of days you are likely to be born on.

Despite this, today seems to be a very popular day for birthdays at Citigate Dewe Rogerson. Fittingly, 20th October is also the seventh anniversary of World Statistics Day, first celebrated in 2010 and again in 2015, and also the seven-year anniversary of the Royal Statistical Society’s statistical literacy campaign (20/10/2010 being an aesthetically-pleasing number) so it’s sort of statistics’ birthday too.

So, happy birthday statistics! (And everyone else celebrating a birthday today)

Written by Chris Jarvis, Head of Research