I’ve always found this to be very annoying, overly simple and unintuitive, but there’s a very interesting result. In a room of 23 people, there is a 50.7% chance that two people share a birthday. Expand this up to 57 people, and there’s a 99% chance that this is true. Conventionally, I’d say you’re one person and there is a n/365 chance that anyone shares your birthday, which is a 5.7% probability. This feels intuitive and comfortable, but as Daniel Kahneman would say to you, “activity recruit your brain to think” and not to allow your fast thinking brain to take the lead. Using your critical, problem solving brain is an active process you have to engage with and try, which is difficult with limited compute time per day. You tend to rest on your laurels with your fast thinking, instructive brain to do most of your day to day tasks which do not pose a hard problem to you. This system also tends to oversimplify and categorise things as easy problems, very over confidently.
So birthday coincidence is a CO-incidence, where you have to consider all of the pairs of people in a room. The number of pairs is:
$$\frac{n(n-1)}{2}$$
Which means in a group of 23 people, there are 253 pairs of people, each person pairing to 22 others. So 253 pairs means 253 chances of an incidence of the same birthday. So what’s the probability that nobody shares a birthday with anyone else? Call that Probability of B, P(B) and the chance that there is a match is P(A).
$$P(A)=1-P(B)$$
To calculate P(B), we need more maths. The number of ways that k people can have 365 distinct birthdays is complex. For one person, you have 365. For two people 365 times 364. For k, Vnr = 365 x 364 x 363 x 362 … x (365 – k) times.
$$V_{nr}=\frac{365!}{(365-k)!}$$
And the total number of ways in which 23 people can have their birthdays arranged, regardless of matching is
$$V_{t}=365^{23}$$
That might be a stretch to wrap your head around if you’re not familiar with factorials. Put more simply, with each person, there is one less day in the year you can have a unique birthday. The first person having a 100% chance, second person having a 364/365 (99.7%) chance, and so on.
$$\frac{V_{nr}}{V_{t}}=\frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \frac{362}{365} \times … \frac{365 – k}{365}$$
And so, the probability is nobody having a match, P(B) is
$$P(B) = \frac{V_{nr}}{V_{t}}=0.492703$$
Which gives the inverse probability of a matching birthday
$$P(A) = 1 – P(B) = 0.507297$$
Wow, we have a 50.7% chance. I hope you enjoyed using factorials, which I’m sure you didn’t expect to do ever again in your life after high school, nor to have so much fun in doing so. Repeat for yourself with more people and see how rapidly it trends towards 100% probability.
What about the chance of three people, you ask? Now you’re just flirting with me. This is also surprisingly high compared to what you may have assumed, previously unarmed with the power of maths. The actual calculation is far too difficult for today, but there are some fellow nerds on Math Exchange who have approximations and for an equivalent 50% likelihood, and it’s 86 people in a room. A surprisingly few number of people to have three matching birthdays.
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