How unlikely was that!?

I was very tempted to call this post 'Spot the Maths/Stats Mistake #5', however there is a big difference between this post and the others I have made so far about the use, misuse and abuse of statistics.  The difference is that this time, the mistake was my own.

On Thursday, I was travelling back from a work event by train.  I didn't realise that I had a reserved seat, so just sat in a seat which was available.  I then discovered that I actually did have a reserved seat, and that I was in fact sat in it.  I thought to myself what an amazing coincidence this was - highly unlikely, I thought.

Cross Country Trains 'Voyager' Seating Plan

However, when I hear that something very unlikely has happened, I try to think about how unlikely it really was.  I genuinely believed that it was highly unlikely that this should happen.  There are 296 seats on this particular type of train, so I thought that the chance of this occurring was 1 in 296.  I was wrong.  Here is why.

• About 70% of the seats were already occupied.
• I didn't have a first class ticket, so 40 of the seats were unavailable to me.
• I sat in the same carriage as a colleague, who had booked his ticket separately to me, but who did have a reserved seat.

So even with those first two observations, the 296 seats I had to choose from were reduced to around 77.  All of the reserved seats were in the front 2 carriages of the train, and since I deliberately sat in the same carriage as my colleague (although I couldn't sit next to him as the seat was occupied) this meant that I selected almost exclusively from the unoccupied reserved seats.  These constituted around half of the total number of unoccupied standard class seats, reducing my choice of seats further, to around 38.  I also know that I booked my tickets at around the same time as my colleague.  Assuming that seats are allocated approximately sequentially (e.g. filling up the train from the front) it was very likely that my reserved seat was in the same carriage as his.  Perhaps this means I had something resembling a 1 in 19 chance of sitting in my own seat.

This is a bit of a rough-and-ready approximation, based on some perhaps unreasonable assumptions and estimates, but I hope it illustrates the central point.  I thought that the probability of me sitting in the right seat was close to 1 in 300.  In fact, it was possibly less than 1 in 20.  Still unlikely, but we often tend to overstate just how unlikely things truly are.