Puzzle time - TNCR piano recital
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The last one was easy so I thought I'd do another.
You are last in a line of 100 people waiting to watch a TNCR piano recital in a venue with 100 seats. Klaus couldn't make it because of Covid flight restrictions, so you do in fact want to attend.
Everyone has an assigned seat, but one of the people in front of you is a Free Spirit who will ignore his ticket and choose a seat at random (which might of course be his own).
The other concertgoers will all obediently follow instructions, except that if one of them finds someone else sitting in his seat he will also take a seat at random.
What are the odds that when everyone is seated you will be in your assigned seat?
You were warned.
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Very small amount.
Have to think about it. depends at what # is the free spirit in line.
Bit still think it is a small number.
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I’ll be sitting in my seat, thank you very much...
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My guess. I'm somewhat confident that it is the right answer, but I don't have a very good rationale for it yet.
||50%. The answer doesn't depend on how many seats/people there are.
For n = 2, the 50% is obvious.
For n = 3, the probability that p1 and p2 sit in their assigned seat is 5/12, and the probability that p1 sits in p2's seat and vice versa is 1/12, which adds up to 50% again.
According to the principle of induction as it applies to engineering, a proposition is true for all numbers if it is true for n=2 and n=3. Qed.
I also noted that the integer sequence A062119 seems to play a role here as the denominator of probabilities of problem size n. For instance, in the n=3 case we talked about numbers with denominator 12, the third number of the sequence. Hmm...
My guess is that the real solution talks about cycles.
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Everyone in front of Free Spirit can be ignored.
When Free Spirit enters, he does one of 3 things:
- Takes his own seat (by chance)
- Takes someone else's seat
- Takes your seat
If the result is #2, the cycle just repeats - the guy whose seat he took essentially becomes the new Free Spirit.
The ulimate outcome depends on either 1 or 3 happening. And 1 and 3 are equally likely.
You were warned.
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MaKes sense.
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Never mind all this Free Spirit nonsense. I think you need to name the asshole who don't do what he's damn well told.
Yeah, I know, it's me.
So, given this case, 1/100.
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Everyone in front of Free Spirit can be ignored.
When Free Spirit enters, he does one of 3 things:
- Takes his own seat (by chance)
- Takes someone else's seat
- Takes your seat
If the result is #2, the cycle just repeats - the guy whose seat he took essentially becomes the new Free Spirit.
The ulimate outcome depends on either 1 or 3 happening. And 1 and 3 are equally likely.
@jon-nyc said in Puzzle time - TNCR piano recital:
Everyone in front of Free Spirit can be ignored.
When Free Spirit enters, he does one of 3 things:
- Takes his own seat (by chance)
- Takes someone else's seat
- Takes your seat
If the result is #2, the cycle just repeats - the guy whose seat he took essentially becomes the new Free Spirit.
The ulimate outcome depends on either 1 or 3 happening. And 1 and 3 are equally likely.
Nice!
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The last one was easy so I thought I'd do another.
You are last in a line of 100 people waiting to watch a TNCR piano recital in a venue with 100 seats. Klaus couldn't make it because of Covid flight restrictions, so you do in fact want to attend.
Everyone has an assigned seat, but one of the people in front of you is a Free Spirit who will ignore his ticket and choose a seat at random (which might of course be his own).
The other concertgoers will all obediently follow instructions, except that if one of them finds someone else sitting in his seat he will also take a seat at random.
What are the odds that when everyone is seated you will be in your assigned seat?
@jon-nyc said in Puzzle time - TNCR piano recital:
The last one was easy so I thought I'd do another.
You are last in a line of 100 people waiting to watch a TNCR piano recital in a venue with 100 seats. Klaus couldn't make it because of Covid flight restrictions, so you do in fact want to attend.
Everyone has an assigned seat, but one of the people in front of you is a Free Spirit who will ignore his ticket and choose a seat at random (which might of course be his own).
The other concertgoers will all obediently follow instructions, except that if one of them finds someone else sitting in his seat he will also take a seat at random.
What are the odds that when everyone is seated you will be in your assigned seat?
I'd just stand up until Ax started having his epileptic fit on the piano, and then I could sit anywhere I wanted...
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@jon-nyc I have a different take on this.
It varies depending on where you are in the order of getting seated. If you are the first in line to get seated, the odds of you remaining in your assigned seat is 100%.
If you are the 2nd in line to get seated, then there is a 1/99 chance that the one person who gets seated ahead of you is the Free Spirit, and that Free Spirit has a 1/100 chance of taking your seat.
Working through all these possible permutations, it is not obvious that the overall odds of you getting your seat is 50%.
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The problem states that you will be the last seated.
Also if the problem were what you are outlining, there would be a 75% chance you’d get your seat. Because even when you are after the Free Spirit, there’s still a 50% chance you’d end up in your own seat.
