Probability of having concert seats 109 and 110 for separate concerts

Question

I went to an Alan Parsons Project concert in Feb 2020. I got seats 109 and 110. I flew from my residence in PA to CA to see Alan Parsons at The Sabin Theater in LA. About Feb of 2020 as well I got tickets to see Ringo Starr in NYC at the Beacon Theater... seats 109 and 110! This was random and only noticed afterwards that seats were the same. Ringo's concert was rescheduled for summer 2022. I know this problem most likely would need the total venue seats, etc. Can one determine the formula for me to use in order to yield the most correct probability if possible or determine what the probability is since it's such a cosmic occurrence? Thank you for your help!

Answer 1

that does seem awfully unlikely!

To answer your question we need to appeal to combinatorics. We're looking for the probability of getting the same two (consecutive!) seats, out of all the possible combinations of seat pairs. But to do this, we definitely need the total number of seats. And to be really precise, we'd need to floor plan to figure out how many adjacent seat pairs there are. A quick google says the theater's capacity is 1,897 which probably includes standing room, but I'm just going to go with that number for now.

First, we want to find the number of combinations of two seats there are. This is given by:

n! / (r!(n-r)!) , where n is the population (ie number of seats) and r is the sample (two, for us because we're looking for pairs of seats. ! is the symbol for factorial)

And when we plug in our numbers, n = 1897 and r=2, we get the result that there are 1798356 possible two-seat pairs.

Here's where it gets a little trickier. Because to find the probability of getting the same two seats at both concerts we don't actually care which two they were the first time, just that we get the same two at the second concert. So we actually just want the probability of getting that one pair of seats at the second concert. But this is just 1/1798356! which is to say it's very very unlikely!

Now if we did look at a floor plan and just include adjacent seats, it would lower the number of possibilities a good bit, both because the total number of seats is certainly lower than the total capacity and we get to exclude all non-adjacent combinations of seats. But the difficulty of getting that answer lies in looking at the floor plan to figure out all the adjacent combos—the math is basically the same.

So that gives us your answer! (or at least a rough answer to a question pretty close to what you are asking!).

Hope that helps!

EDIT: I just re-read your question and realized you're talking about two different theaters, not the same seats at the same theater! sorry about that. But I can take a swing at that too if you'd like