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### Hard Logic Probability Puzzle

Hard Logic Probability Puzzle - 11 July

Bruna was first to arrive at a 100 seat theater.
She forgot her seat number and picks a random seat for herself.

After this, every single person who get to the theater sits on his seat if its available else chooses any available seat at random.
Neymar is last to enter the theater and 99 seats w
ere occupied.

Whats the probability what Neymar gets to sit in his own seat ?

1. I say 50%.
In the trivial case where there are only 2 seats, its a 50/50 whether Bruna takes Neymar's seat.
If there are 3 seats, there are 3 possibilities:
1) Bruna sits in her own seat - in which case Neymar will for sure get his own seat.
2) Bruna sits in Neymar's seat - in which case he for sure doesn't get his seat.
3) Bruna sits in the 2nd person's seat. When person 2 arrives, it's 50/50 if he takes Neymar's seat.
Thus the probability is 1/3 + 1/3*1/2 = 50%.
This will be true for any number of seats.

1. This comment has been removed by the author.

2. 1st 2 points are right. I don't think the last point is.

When 2nd person arrives, his prob. of sitting on Neymars seat is 1/99 not 1/2.

Also, if that 2nd person arrives after 49 other have arrived for ex, then his prob of sitting in Neymars seat is 1/50.

Point being, I don't know the answer, but it cant be 1/2

3. for 4 people: 1/4 + 2/4*(1/3+1/3*(1/2)) = 0.5

It seems you are correct...
This sequence will return 0.5 allways :)

2. I got expression for probability for any persons to get their respective right seat. Curious whether brain teaser is interested to know it. For the last person it will be 1/2 irrespective of no. of seats in the problem.

The administration solution based on the two possibilities is not clear unles they provide arguments that both are equally probable.

3. the answer lies on 2 cases:-
case 1: if bruna sits in her own seat.
prob. that Neymar gets his own seat = '1'
(as question says that after neymar each person entering the theatre sits in his/her own seat.)
case 2: if bruna doesn't sit in her own seat.
prob. that Neymar gets his own seat = '1/2'
hence there would be two answer for this ques.(1,1/2)

4. it wii be 100 : 1

5. So if there are two possibilities the probability must be 1/2?

You cannot be more wrong.

proof 1: the probability that he gets his seat is if his seat was not taken by the previous 99 people.

the probability string of his seat not being taken is 99/100 * 98/99 * 97/98 .. ...... which evaluates to 1/100.

proof 2: the probability of any seat being left out by the first 99 people are equally likely and mutually exclusive.

hence, the probability of his seat being the one that's left out is 1/100.

1. retarded as fuck...

2. I agree, I was, I apologize for that behavior, I was reading my own comment without realizing that it was my own comment and I fucking hated him not only because he was wrong, but he shitty attitude that he had as if he's better than everyone.

I can't express how shameful I am that it was me.

Go ahead world, please all tell me how much garbage I am.

But no seriously OP, Sorry, So sorry for such behavior and being wrong and calling you wrong before thinking things through.

6. Assuming everyone else comes in one by one instead of all at once...

case 1 Bruna got the right seat (1%)
case 2 Bruna got the wrong seat (99%)

Neymar got his seat given case 1 = 100%
Neymar got his seat given case 2 = 98!/99!=1.01%

Total:
(case 1)1% x 100% + (case 2) 99% x 1.01% = 2%