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The Birthday Paradox
One of our data scientists, Ruth Boardman, discusses the probability of two people sharing a birthday.

The birthday paradox answers the question: if we have x number of people what is the probability at least two of them share a birthday?

With 365 days of the year the temptation is to think that sharing a birthday with someone would be a rare event. However the maths says otherwise. I have plotted the probability of two people sharing the same birthday in a group size ranging from two to sixty.


Once the group size reaches 23 there is a greater than 50% chance that two of the group members share a birthday, by 56 group members there is a 99% chance of two sharing a birthday. This seems to defy intuition!


The brain loves to consider problems from its own perspective and ends up translating the birthday question into ‘who shares a birthday with me?’ when it is really ‘who shares a birthday with anybody?’ Falling into this trap dwarfs the number of comparisons we have available to us. Take 56 people in a room, if only comparing yourself to everyone you have 55 comparisons to make, if you compare everybody with everybody you have 1,540 chances to match a birthday. Much better odds!

The other aspect to this paradox is how quickly exponents escalate. It is actually easier to calculate the scenario where no one shares a birthday. The chance of two people having different birthdays is:


364/365 = 0.997


Which means we have a 1 - 0.997 = 0.003 chance of a match. That is for just one comparison. We have 1,540 to make, we can multiply the fraction by itself for each comparison we make. Ie:


364^1540/365 = 0.014


Which leaves us with 1 - 0.014 = 0.986 chance of two people sharing a birthday. The power of exponents (pun intended) made our big number 0.997 into a small number with the addition of 54 people.

The green vertical line on the plot marks the odds at 26 people in the room. 26 is a special number here at Fospha as there are 26 of us. According to the maths we have a 60% chance of two team members sharing a birthday. And we do have a match in the team!