# Just curious of the odds of this happening

Was having lunch with a lover on the boardwalk and he pulled a random couple off of the boardwalk to have lunch with us. As we were talking, we discovered that both men had the same birthday, and the woman shared her birthday with me. (Not year; just dates). We pulled out our ids to prove this to each other. What are the odds of this happening? (It was also weird that she and I shared the same first and middle name, albeit spelled differently, but that's not relevant to this analysis.) TLDR: What are the odds that a random couple you meet will share birthdays? What are the odds that the birthdays will specifically match by gender?

• Probability calculations relate to events specified before the observation. This problem arises in various guises in statistics (it's equivalent to getting a hypothesis from data and then using the same data to test the hypothesis). See here which gives an example that highlights the sort of silly answers we get if we do that. If you see something weird happen, and go "Wow, what are the odds of that?" you're in effect saying "what are the odds that something surprising enough for me to ask about would happen in some period of time",.. (ctd) – Glen_b May 17 '15 at 1:18
• ... (ctd) On that basis both answers really address a calculation for a situation different from the one you're in. If you'd said, just as your companion was first talking to the other couple "I wonder what the chances are that she'll share my birthday and the two men will as well" ... the answers attempt to address that question. If that happened, it would be astonishing. But when noticing something 'surprising' after the fact, there are so many other surprising things that might occur (e.g. the woman grew up in the next street from your companion) --> "different numbers on the dice". – Glen_b May 17 '15 at 1:27

Let m1 and f1 denote your birthdays (values between 1 and 365). Assume you have distinct birthdays. For the sake of simplicity and without loss of generality you might assume m1=1 and f1=2. Let m2 and f2 denote the birthdays for the other couple.

The probability of sharing the birthdays with the other couple is simply the probability of (m2=1, f2=2) or (m2=2, f2=1) which is equal to 2/365^2 hence

What are the odds that a random couple you meet will share birthdays?

the odds is $\Large\frac{\frac{2}{365^2}}{\frac{365^2 \,-\,2}{365^2}}=\frac{2}{365^2\,-\,2}$ .

What are the odds that the birthdays will specifically match by gender?

Similarly is $\Large\frac{1}{365^2\,-\,1}$

• Just to finish the calculation: Odds of sharing birthdays is 1 in 66611.5 or 2 in 133223. Odds of also matching gender is 1 in 133224 – Eric Farng May 16 '15 at 21:47

Assuming all days of the year are equally likely for a birthday, and accounting for the leap year, then there is a $1/365.25$ probability of being born on a given day. Before answering your question, note that the probability of two people sharing a birthday is $1/365.25$.

You want to know the probability of this happening twice (male birthday same as other male birthday and female birthday same as other female). The probability of this is $(1/365.25) * (1/365.25)$, which is roughly $0.0000075$ or one in $133,408$. In other words there are $133,408$ possible combinations of birthdays for two people. If you grabbed a random couple every day at lunch, you would expect this event to happen once every $365.25$ years.

For the second question, note that the probability of your birthday matching either the other female or the other male is $1/365.25 + 1/365.25$. The probability of this happening twice is $(1/365.25 + 1/365.25) * (1/365.25 + 1/365.25)$ which simplifies to $4/365.25^2$. This is roughly $0.00003$ or one in $33,351$. This would happen roughly once every $91$ years.

For such small numbers the odds are approximately equal to the probabilities.