One hundred people board a 100-seat airplane. The first one has lost

Question

One hundred people board a 100-seat airplane. The first one has lost his boarding pass, so he sits in a random seat. Each subsequent passenger sits in his or her own seat if it’s available or takes a random unoccupied seat if it’s not. What’s the probability that the 100th passenger finds her seat occupied?

A. 0.2
B. 1
C. 0.034
D. 0.5
E. 0.1

A. 0.2
B. 1
C. 0.034
D. 0.5
E. 0.1

Lucky2783 · Answer

Lets try to find the probability that 100th person finds the seat not occupied . 
100th seat will not be occupied when second person who comes after first person sits on First Person's seat , there is 50% chance that the new passenger will board First Person's seat and so 50% chance that he will not board the first person's seat. 
so the probability of 100th person finding the seat unoccupied is 50% . hence there are 50% chances that the 100th person seat is occupied . 
Answer D .

MarkusKarl · Answer

When the first person comes into the airplane he can choose 100 different seats, only one of the outcomes of his choice would result in the case that the last person gets to sit at his own seat. As of such there is a 1/100 probability that the last person will have find his or her seat free, and 99/100 probability that he or she will not. But 1% was not provided as an answer choice and 50% was given as the correct answer.

The reasoning behind this question, if 50% were to be true, is simply comparable to a coin flip(taken or not taken). That's not really fair imo as we would ignore the entire universe of different scenarios and just look at the last step.

What am I missing?

CrackverbalGMAT · Answer

Answer is D which is equal to 0.5.
Lets take this questions instead of 100 seat airplane. Lets take it as four seat airplane.
Chances of fourth person seat will not be occupied depends on the 2 nd person who comes after the first person.
If the 2nd person who comes after the first person sits in the first persons seat then the fourth person seat will not be occupied.
So there is 50% chance that fourth person seat is not occupied.
So there is 50% chance that fourth person seat is occupied.
Similarly for the 100 seat airplane.
So answer is D.

tronghieu1987 · Answer

I am confused of this explanation.
- Firstly: why the chance that the 2nd person takes the first one's seat is 50%. If 3 seats are left, I think the chance must be 33%.
- Secondly: Make the above chance of 50% is right. But assume that the 2nd person doesn't take the first person's seat, rather takes the 3rd person's seat. Then if the 3rd one takes the 1st one's, then the 4th one will still find his/her seat available. Thus, I think a stop at the 2nd person is not enough.

I take the even more simple scenario and then could find the answer which is 0.5. Make it just 3-seat-plane.
- 1/3 is the chance that the 1st person choose his/her seat correctly =>the last one (3rd) will find his/her seat available;
- 1/3 the chance the 1st one takes 2nd one's seat. Here, 1/2 chance the 2nd takes the 1st one's, then 3rd one's is available; OR 1/2 chance the 2nd takes the 3rd one's, then 3rd one's is not available;
- 1/3 the chance the 1st one takes 3rd one's, then the 3rd one's is not available.
So the chance the 3rd person's seat is not available is: 1/3*1/2 + 1/3 = 1/6 + 1/3 = 0.5. (D).

I just don't know is it the same to the 100-seat-plane. Need a much more simple way. Thanks

shriramvelamuri · Answer

I tried to solve this question from the answer choice.

To me the central logic is..,

No matter how many passengers and how many passengers are available, the only two option for a seat is either 1. occupied or 2. Vacant. (lets see the question from the seat perspective and not from the person perspective)

So there is always a 50% chance for every seat in the plane.

Please correct me if wrong.

Lucky2783 · Answer

regarding your first doubt :-  Firstly: why the chance that the 2nd person takes the first one's seat is 50%. If 3 seats are left, I think the chance must be 33%. 
Let me try to help , we are only concerned about probability that an event happens or it doesn't happen , the next passenger takes first passenger's seat or he does not take the first passenger's seat . it is an independent event and is not affected by what he choose if he does not choose first's person seat. 
please have a read on independent events , it is an important concept on GMAT exam.

tronghieu1987 · Answer

I do understand the independent event concept.
However, this case is not an independent event. We knew that there are 4 seats, and the first person took the 2nd seat. Thus, there are 3 seats left, including the 1st seat. So the chance that the 2nd person could take the 1st seat is 1/3, and the chance that he/she couldn't is 2/3.

Maybe you misunderstand my question, so in turn I misunderstand your explanation:).

Posted from my mobile device

Bunuel · Answer

Note that this is not a GMAT question. Topic locked.

One hundred people board a 100-seat airplane. The first one has lost

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