Monday Morning Braintease: airplane seats

There are 100 people in line to board an airplane. Each person is holding a ticket corresponding to both his place in line and his seat (i.e. Person 1 is supposed to sit in Seat 1, Person 2 is supposed to sit in Seat 2, etc.). All of the people in line are normal, except for Person 1, who is an idiot. When Person 1 boards the plane, he will ignore his seat number and sit in a random seat. Following Person 1, each person will sit in his own seat, unless his seat is occupied. In this case, the Person who's seat is occupied becomes the new idiot and will sit in a random seat. For example, if Person 1 sat in Seat 2, Person 2 will become the idiot and sit in a random seat. What is the probability that Person 100 will end up in Seat 100?

 

Are we assuming a uniform distribution over the seats for the idiot? If so, this is pretty simple. 50%.

Not hard to prove rigorously, but just look at the base cases of 2, 3 and 4 people to convince yourself.

-MBP
 

yea 50%....50% chance the guy sits in his seat or not...but based on the fact that he's an idiot, i highly doubt he'll walk all the way to the back of the plane without falling down or tripping

I eat success for breakfast...with skim milk
 
Best Response
bob_bsd:
Sorry, can someone walk me through it? I feel like I'm over-thinking it.
Assuming a uniform distribution over the seats for the idiot.

If there are only two passengers, it's the chance that person 2 ends up in seat 2 is 50%, because this is the chance person 1 will randomly choose seat 1.

If there are three passengers, the chance the person 1 chooses seat 1 is 1/3, and the chance that person 1 chooses seat 2 is 1/3. If person 1 chooses seat 2, the chance that person 2 chooses seat 1 is 50%. So the chance neither person 1 nor 2 end up in seat 3 is 1/3 + 1/3*1/2) = 0.5.

If there are four passengers, the chance that person 1 chooses seat 1 is 1/4. The chance he chooses seat 2 is 1/4. If he chooses seat 2, person 2's chances of choosing seat 1 are 1/3 and choosing seat 3 are 1/3. If he chooses seat 3, person 3s chances of choosing seat 1 are 1/2. If person 1 chooses seat 3, person 2 will choose seat 2, and person 3 has a 1/2 chance of choosing seat 1. Add up all these probabilities and you get 50%.

There is a similar analogy (which gets more and more cumbersome as the number of passengers goes up) for every number of passengers.

-MBP
 
Buddyfox:
Not sure why I still cant wrap my head around this. Looks like I'm the idiot on this plane.
Well, do you get the base case examples I pointed out? The clearest one is the example with 3 people (the 2 person case is trivial).

Let's call P(x,y) the probability that person x ends up in seat y and P( (x,z) | (x-1,y) ) to be the conditional probability that person x will end up in seat z given person x-1 ends up in seat y. i.e. P( (2,1) | (1,2) ) is the probability that person 2 ends up in seat 1 given person 1 picks seat 2.

So the three person case is:

P(3,3) = P(1,1) + P( (2,1) | (1,2) ) = 1/3 + 1/21/3 = 1/3(1 + 1/2) = 1/2

The four person case is:

P(4,4) = P(1,1) + P( (2,1) | (1,2) ) + P( (3,1) | (1,3) ) + P( (3,1) | (2,3) ) = 1/4 + 1/41/3 + 1/41/2 + 1/41/31/2 = 1/2

And so on...

-MBP
 
manbearpig:
Buddyfox:
Not sure why I still cant wrap my head around this. Looks like I'm the idiot on this plane.
Well, do you get the base case examples I pointed out? The clearest one is the example with 3 people (the 2 person case is trivial).

Let's call P(x,y) the probability that person x ends up in seat y and P( (x,z) | (x-1,y) ) to be the conditional probability that person x will end up in seat z given person x-1 ends up in seat y. i.e. P( (2,1) | (1,2) ) is the probability that person 2 ends up in seat 1 given person 1 picks seat 2.

So the three person case is:

P(3,3) = P(1,1) + P( (2,1) | (1,2) ) = 1/3 + 1/21/3 = 1/3(1 + 1/2) = 1/2

The four person case is:

P(4,4) = P(1,1) + P( (2,1) | (1,2) ) + P( (3,1) | (1,3) ) + P( (3,1) | (2,3) ) = 1/4 + 1/41/3 + 1/41/2 + 1/41/31/2 = 1/2

And so on...

Degree & Major: math

Haha, figures. I wish I would have studied math to become a critical thinker like you. Kudos, bro!

The difference between successful people and others is largely a habit - a controlled habit of doing every task better, faster and more efficiently.
 

MBP how did you get so good at math

manbearpig:
Buddyfox:
Not sure why I still cant wrap my head around this. Looks like I'm the idiot on this plane.
Well, do you get the base case examples I pointed out? The clearest one is the example with 3 people (the 2 person case is trivial).

Let's call P(x,y) the probability that person x ends up in seat y and P( (x,z) | (x-1,y) ) to be the conditional probability that person x will end up in seat z given person x-1 ends up in seat y. i.e. P( (2,1) | (1,2) ) is the probability that person 2 ends up in seat 1 given person 1 picks seat 2.

So the three person case is:

P(3,3) = P(1,1) + P( (2,1) | (1,2) ) = 1/3 + 1/21/3 = 1/3(1 + 1/2) = 1/2

The four person case is:

P(4,4) = P(1,1) + P( (2,1) | (1,2) ) + P( (3,1) | (1,3) ) + P( (3,1) | (2,3) ) = 1/4 + 1/41/3 + 1/41/2 + 1/41/31/2 = 1/2

And so on...

 

The probability is 100%!! Because person 1 is such an idiot that he boarded a train instead of an airplane. This means when person 2 boards the plane is empty and all passengers will be seated correctly.

 
pdoher01:
The probability is 100%!! Because person 1 is such an idiot that he boarded a train instead of an airplane. This means when person 2 boards the plane is empty and all passengers will be seated correctly.

Hahahhahaha omg. Great attention to detail.

 
pdoher01:
The probability is 100%!! Because person 1 is such an idiot that he boarded a train instead of an airplane. This means when person 2 boards the plane is empty and all passengers will be seated correctly.
LOL i thought the exact same thing but just figured it was an error :P
"If you survive to my age and you rack up a CV like mine, you can look at HR and say, "Fuck you. I don't try out."- Eddie
 
turtle:
pdoher01:
The probability is 100%!! Because person 1 is such an idiot that he boarded a train instead of an airplane. This means when person 2 boards the plane is empty and all passengers will be seated correctly.
LOL i thought the exact same thing but just figured it was an error :P
Amusing answer, but the question clearly mentions that 100 people are in line to board the airplane, and the first person in line is the idiot.
-MBP
 

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