Sunday, August 3, 2014

The Birthday paradox at a wedding

I know I've been very quiet of late but I started a new job 4 months ago and have been pouring myself into it to learn as much as I can. More importantly I've been preparing for my wedding to the lovely Lelly Ann, though to be honest she has been doing most of the tough work so she deserves all the glory!.

The big day is only 5 days away now. I had great hopes of writing a cool optimizer that would seat guests based on their affinity but unfortunately laziness and an over-estimation of my programming skills got in the way. However I have been thinking about the birthday paradox recently and since there will be over 100 people in the room next Friday I thought it was a nice anecdote.

The Birthday paradox arises from the chances of two or more people in a group having the same birthday. Given that there are 365 days (ignoring leap years for simplicitys sake) in a year you would think that the chances that any 2 people might have the same birthday would be extremely low;
however this is not the case.
Wikipedia has a great page on it so I won't reproduce their excellent explanation but it turns out that at 23 people the odds tip over 50%, which is better than a coin toss. In terms of our day we are due to have 106 guests.
The formula is:
1 - (Permutation(365,n)/(365^n)) Where n is the number of people involved.

so 106 guests works out at 99.99999574936430000% or as close to 100% as makes no difference.
We also have tables of 8,10 & 11 which work out respectively at 7.43%, 9.46% & 14.11% respectively. I wonder would it be worth sampling each table to see how many times this actually comes through. with 10 tables we should probably see this at least once!