Friday Puzzler--100-seat plane

Re: Friday Puzzler--100-seat plane

Larry Barrett

Suppose it is a 2 passenger plane.

A boards and could sit randomly in seat A (prob = 1/2) or seat B (prob = 1/2). So the probability that B sits in the correct seat is 1/2.

Now suppose it is a 3 passenger plane.

A boards and could sit randomly in seat A (prob = 1/3), seat B (prob = 1/3) or seat C (prob = 1/3).

If A sits in seat A, then both B and C will sit in the correct seats. If A sits in seat B, then the prob that B sits in seat A is 1/2 and then C will sit in the correct seat.

So the probability that C sits in the correct seat is 1/3 + 1/3 x 1/2 = 1/3 + 1/6 = 1/2.

It is an exercise for the reader to work out the probability for a 4 passenger plane.

...

So my guess is that in a 100 seat plane, the probability that the 100th passenger sits in the correct seat is 1/2.

Correct!

Alex Y

One way to see the answer (no matter how big the plane) is to give the first person a distinctive blue hat. Whenever someone sees their seat is taken the person in their seat hands them the blue hat to wear, and they pick another seat at random.

The answer for the last passenger is determined when the wearer of the blue hat either sits in the first passenger's assigned seat or in the last passenger's seat, and those possibilities are equally likely each time a person in a blue hat chooses a seat.