urgent math help

Question

the question is -  A fully booked theatre is about to screen A Beautiful Mind to a group of 100 mathemati- cians. the 1st mathematican was thinking about the Riemann Hypothesis and somehow misplaced his ticket. As a consequence, he could not find the seat alloted to him and decided to sit at a random seat. From then on, any mathematician if he finds his seat oc- cupied, politely goes and seats at a random seat. What is the probability that the 100th mathematician seats in his own seat?

Answer 1

Funny but interesting question.

 

The key to this is to realize that two events are possible

 

1)Riemann guy sits in his correct seat

2)Riemann guy does not sit in his correct seat

 

P1=1/100

P2=99/100

P100: Probability that 100th guy gets correct seat

 

If(P1), then P100=100% since everyone finds his correct seat

 

If(P2), then P100=1/99 since it will be a random chance that P100 will get his correct seat.

 

Using conditional probability

 

P100=P1*1 + P2*(1/99)=.01 + 99/100*1/99)=0.02¹

P100=2%

 

1: I'm not 100% confident with this, but it feels correct.  Even if Riemann guy fails, there is still a chance that the others will fail to find the 100th guys' seat.

 

Update!

 

This has been bugging me all day.  I think the answer is closer to 50% after further thought.  I started with 2 seats and added an additional seat to work through the sample spaces.  It is not uniformly 50% independent of #seats, but approaches this answer asymptotically.  Still working on it ;-)   Nice problem!!

 

Final Answer: I'm convinced - the answer is 50% regardless of how many seats - excellent problem.