Timeline for Taking Seats on a Plane
Current License: CC BY-SA 4.0
26 events
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| Sep 1, 2021 at 15:17 | history | edited | Will Orrick | CC BY-SA 4.0 |
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| Sep 1, 2021 at 15:10 | history | edited | Will Orrick | CC BY-SA 4.0 |
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| Sep 1, 2021 at 12:53 | comment | added | Will Orrick | In the sum $\sum_{k=2}^{n-1}p_{n-k+1}$, the subscript of $p$ goes down in steps of $1$ as $k$ goes up in steps of $1$ (because of the $-k$ in $n-k+1$). The subscript of the first term is $n-2+1=n-1$, which decreases in steps of $1$, reaching $n-(n-1)+1=2$ in the final term. So the reversed sum is $\sum_{k=2}^{n-1}p_k$. | |
| Sep 1, 2021 at 7:54 | comment | added | user851668 | 10. Pls expatiate $\Pr(\text{$N$ gets their assigned seat})=\sum_{k=1}^{N-1}\Pr(\text{$k$ sits in $1$}\mid E_k)\Pr(E_k)=\frac{1}{2}\sum_{k=1}^{N-1}\Pr(E_k)=\frac{1}{2}$ to a 15 y.o.? How can something as simple as $\Pr(\text{$N$ gets their assigned seat})$ equal something as abstruse and Gordian as $\Pr(\text{$N$ gets their assigned seat})=\sum_{k=1}^{N-1}\Pr(\text{$k$ sits in $1$}\mid E_k)\Pr(E_k)=\frac{1}{2}\sum_{k=1}^{N-1}\Pr(E_k)=\frac{1}{2}$? The LHS is a number, but the RHS has Capital Sigma Notation! LHS is a probability, but RHS a Conditional Probability! See why I'm wildered? | |
| Sep 1, 2021 at 7:51 | comment | added | user851668 | Many thanks! Without "Writing out the summation explicitly", how can you intuit that $\sum_{k=2}^{n-1} p_{n-k+1} \equiv \sum_{k=2}^{n-1}p_k$? | |
| Aug 31, 2021 at 13:28 | comment | added | Will Orrick | 7. After multiplying by $n$ the equation is $np_n=1+\sum_{k=2}^{n-1} p_{n-k+1}$. Writing out the summation explicitly, one gets $p_{n-1}+p_{n-2}+\ldots+p_3+p_2$. Reversing the order gives $p_2+p_3+\ldots+p_{n-2}+p_{n-1}$, which, as a summation, is $\sum_{k=2}^{n-1}p_k$. Hence the equation becomes $np_n=1+\sum_{k=2}^{n-1}p_k$. | |
| Aug 31, 2021 at 13:17 | comment | added | Will Orrick | 5. When it comes time for the $k$th passenger in line to take their seat, $k-1$ passengers will already be seated. Therefore there will be $N-(k-1)=N+1-k$ seats available. For example, passenger $1$ has $100$ seats to choose from, which is $101-1$, not $99$ seats to choose from, which is $100-1$. 6. There are $N$ seats on the plane, but there are also $N$ passengers in line. I have numbered passengers according to their position in line. So there is a passenger numbered $N$. | |
| Aug 31, 2021 at 13:11 | comment | added | Will Orrick | 1. displaced = "displaced from their seat because another passenger has sat there" not "displaced from the line" 8. Nobody suddenly shows up anywhere. If the line contains a displaced passenger, that person could be in any position in line, including the head of the line. | |
| Aug 31, 2021 at 7:17 | comment | added | user851668 | 1. What do you mean by "somebody in line is displaced"? Nobody's forcing passengers out of the queue! Every successive passenger must sit down one of the N seats! 8. Why would a "displaced passenger" suddenly show up "at the head of the line"? | |
| Aug 31, 2021 at 7:13 | comment | added | user851668 | Thanks! 5. "In general, when displaced passenger k chooses a seat, they choose uniformly at random from 101−k possible seats." Where does $101 = n+1$ spring from? Why isn't this $100=n$? After all, there are merely $100=n$ passengers. 6. "$\Pr(\text{$N$ gets their assigned seat})$" Did you mistype this? You defined $N$ as the seat numbers, correct? How can $N$ get their assigned seat? 7. How does $\sum_{k=2}^{n-1}p_{n-k+1} \equiv np_n=1+\sum_{k=2}^{n-1}p_k.$? How exactly did you reverse "the order of summation"? | |
| Aug 25, 2021 at 13:28 | comment | added | Will Orrick | ...to try to eliminate variables by looking for cancelations. Subtracting successive equations cancels a lot of variables, so it's worth trying. 1. What troubles you about the claim? Is it that at most one of the standing passengers is displaced? Or is that all standing passengers have the same probability of being displaced? | |
| Aug 25, 2021 at 13:27 | comment | added | Will Orrick | In reverse order: 4. This is explained in the answer, "In general, when displaced passenger $k$ chooses a seat, they choose uniformly at random from $101-k$ possible seats." (Replace $101$ with $n+1$.) 3. What do you know about conditional probability? Most 9 year olds don't know a thing about it, so it may be hard to come up with a suitable explanation. Do you agree that the events $E_k$ with $1\le k\le N-1$ are mutually exclusive and that one of them must occur? 2. You have a system of linear equations, $2p_2=1$, $3p_3=1+p_2$, $4p_4=1+p_2+p_3$, $5p_5=1+p_2+p_3+p_4,\ldots$ It is natural... | |
| S Aug 24, 2021 at 5:53 | history | edited | Arctic Char | CC BY-SA 4.0 |
Will Orrick forgot to link to that poster's answer.
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| S Aug 24, 2021 at 5:53 | history | suggested | user53259 | CC BY-SA 4.0 |
Will Orrick forgot to link to that poster's answer.
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| Aug 24, 2021 at 5:39 | review | Suggested edits | |||
| S Aug 24, 2021 at 5:53 | |||||
| Aug 24, 2021 at 4:13 | history | edited | Arctic Char | CC BY-SA 4.0 |
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| Aug 24, 2021 at 4:11 | history | rollback | Arctic Char |
Rollback to Revision 3
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| Aug 23, 2021 at 13:42 | history | rollback | robjohn♦ |
Rollback to Revision 2
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| S Aug 23, 2021 at 8:34 | history | suggested | user851668 | CC BY-SA 4.0 |
Directly linked to the other answers mentioned on this page.
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| Aug 23, 2021 at 8:15 | comment | added | user851668 | 4. "The probability of occurrence of these two cycles is the same: $\frac{1}{100}\frac{1}{55}\frac{1}{48}\frac{1}{26}\frac{1}{13}$". The denominator in these fractions is $n - k + 1$, correct? But where did $\frac{1}{n - k +1}$ hail from? This appears to come out of left field! | |
| Aug 23, 2021 at 8:12 | comment | added | user851668 | 3. Pls explain like I'm 9 year old "$\Pr(\text{$N$ gets their assigned seat})=\sum_{k=1}^{N-1}\Pr(\text{$k$ sits in $1$}\mid E_k)\Pr(E_k)$". More details pls! | |
| Aug 23, 2021 at 8:10 | comment | added | user851668 | 2. Pls expatiate and naturalize "Subtracting this from the same recurrence with n replaced by n+1 gives $(n+1)p_{n+1}-np_n=p_n,$ for $n\ge 2$. How did you prognosticate this subtraction? Where did this fey, sibylline step come from? This step appears to come out of left field! | |
| Aug 23, 2021 at 8:07 | comment | added | user851668 | 1. Can you pls expatiate this sentence? Pls explain like I'm 9 years old. Pls edit your answer to clarify. You wrote "If n passengers are standing in line and somebody in line is displaced—it will always be exactly one passenger—the probability that the displaced passenger is at the head of the line is, by symmetry among the standing passengers, $1/n$". | |
| Aug 23, 2021 at 8:01 | review | Suggested edits | |||
| S Aug 23, 2021 at 8:34 | |||||
| Jul 28, 2021 at 16:12 | history | edited | Will Orrick | CC BY-SA 4.0 |
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| Jul 28, 2021 at 4:02 | history | answered | Will Orrick | CC BY-SA 4.0 |