The Boarding Pass Problem: A Scramble for Seats

A previous post in this blog tackles the Birthday Paradox, showing how our minds fail to grasp the probability of real-world events. The missing boarding pass problem embodies a similar concept in a very different way. The problem is most often asked in the form of a riddle: You’re standing in a queue, waiting to board your flight. The first passenger has lost their boarding pass; because of this, they take a random seat. The line consists of 100 passengers, and each subsequent passenger will sit in their assigned seat if it’s available or choose randomly if it’s occupied. What is the probability that you, the last passenger in line, will be able to sit in your assigned seat?

At first glance, the riddle may seem perplexing, but let’s break it down step by step. As the first passenger loses their boarding pass, they are assigned a random seat. For the next passenger, there are two possibilities: either they sit in their assigned seat (the first passenger’s seat) or they select a random seat. If they sit in their assigned seat, the chain continues with each subsequent passenger smoothly occupying their designated seat. In this scenario, you, the last passenger, will have your assigned seat available.

Now, let’s consider the second possibility. If the second passenger chooses a random seat instead of occupying the first passenger’s seat, a chain reaction occurs. Each subsequent passenger will face the same choice: sit in their assigned seat or select a random one if it’s occupied. Eventually, it will come down to the last passenger in line.

Experimental trials with smaller numbers like 3,4, and 5 passengers all result in an answer of 1/2, leading us to believe that this is not a coincidence. In fact, the answer is 1/2 even in the case of 100 passengers. This can be revealed when we make the claim that when the last passenger enters the plane, the seats of passengers 2-99 will be occupied without exception. This is because each passenger would have sat in that seat if it was not occupied, so there must have been someone already sitting there.

Thus, the last passenger is left with either seat #1 or 100. When we look at the context of the problem, the two seats are indistinguishable from the passengers. That is, there is no reason that either one should be occupied before the other for any reason. That means their probabilities of occupation are equal, and so there is a 1/2 chance that each seat is the last one left.

Therefore, in the missing boarding pass math riddle, the probability of the last passenger being able to sit in their assigned seat is 1/2 or 50%. This probability seems quite misleading given that there are 100 seats in the aircraft. But the boarding pass problem illustrates that even in a system where chaos is introduced, the world has a way of stabilizing itself at the end.

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