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Chapter 5: Problem 24

Six individuals, including \(A\) and \(B\), take seats around a circular table in a completely random fashion. Suppose the seats are numbered \(1, \ldots, 6\). Let \(X=\) A's seat number and \(Y=\) B's seat number. If A sends a written message around the table to \(\mathrm{B}\) in the direction in which they are closest, how many individuals (including A and B) would you expect to handle the message?

Short Answer

Expert verified
3 people would be expected to handle the message.

Step by step solution

01

Understanding the Problem

We have 6 people sitting in a circle, each in a randomly numbered seat from 1 to 6. We are interested in finding the expected number of people involved when person A sends a message to person B through the shortest route in the circle.
02

Define Possible Routes

For a circular arrangement, there are two possible routes the message from A to B can take: clockwise or counterclockwise. Our interest is in determining for each seat arrangement, which route involves fewer people.
03

Calculate Distances

Regardless of A's position, compute the clockwise and counterclockwise distances between A and B. For 6 seats, when A sits at seat 1 and B is at seat i, the clockwise distance is \(i-1\) and counterclockwise is \(6-(i-1)\).
04

Determine Effective Path

For every possible arrangement (1 through 6) of B relative to A, determine the path that results in fewer people handling the message. This ensures determining the actual routing based on minimal distance.
05

Compute Expected Involvement

Subdivide the circle into three types based on minimum distance outcomes: same sector (A to B), through exactly one adjacent (A, someone else, B), or full circle. Given uniform distribution and symmetry: - In \((1,1)\): only A and B. - \((1,2)\) or \((2,1)\): A, one person, and B. - (Overlap leads to up to 3 max).
06

Determine Probabilities

Calculate the probabilities for each distinct type of message chain using proportions of \(\frac{1}{3}\) for each unique configuration (direct, one adjacent, 3 adjacent paths) because they exhibit symmetry.
07

Calculate Expected Value

Compute the expected value for the number of people involved. This is the sum of the product of the path lengths by their probabilities:\[ E = 2 \, (\frac{1}{3}) + 3 \, (\frac{2}{3}) = \frac{4}{3} + 2 = \frac{10}{3} = 3.33 \approx 3 \text{ people} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability fundamentally concerns how likely an event is to occur. In this context, we are examining the likelihood that a particular sequence of events, or seat arrangements, will take place.
When six individuals are seated at a circular table, the arrangement of seats impacts the probability of a message taking a particular path. Every unique seat configuration ensures different paths that the message could theoretically follow from Person A to Person B.
  • The total number of distinct seat arrangements is given by permutations, considering rotations make some arrangements indistinguishable.
  • This means the calculation is different from a linear arrangement; hence, circular permutation arises here.
  • There are two potential paths鈥攃lockwise and counterclockwise鈥攆or delivering the message, influenced by probability as a measure of their respective occurrences.
Understanding circular systems with equal distribution remarkably simplifies the probability modeling, allowing us to predict seating outcomes efficiently.
Expected Value
Expected value is a key concept depending on probability, often used to find the average outcome of a random event. In this problem, we want to calculate the average number of people involved when a message is sent around the table by A to B through the shortest path.
We calculate expected value by considering every possible path the message could take and weighting it by its probability.
  • The expected value here reflects the typical number of people that will be involved in delivering the message.
  • To find the expected value, add the products of paths' lengths by their respective probabilities.
  • Each circle part lends itself to a simple calculation due to symmetry, where the expected value aids in summarizing diverse seating arrangements into an understandable mean outcome.
Thus, understanding expected value gives a practical insight into random occurrences, making it a powerful tool for anticipating outcomes of circular seat arrangements.
Seat Arrangement
Seat arrangement in circular permutations significantly impacts the paths that are possible when moving around a table. In circular permutations, each position affects its neighbors, introducing unique considerations compared to linear ones.
  • Circular permutations result in fewer unique positions because rotations of the same sequence are considered identical. This changes the typical factorial calculation for permutations.
  • Each person's seat number pertains to how paths develop 鈥� whether they are direct, involve one adjacent person, or encompass half the circle through three others.
  • Analyzing every possible configuration systematically unfolds the best route either clockwise or counterclockwise for A to send the message to B.
Understanding seat arrangement ultimately helps in determining how a seemingly simple task is influenced by the presence and arrangement of individuals around the round table.
Random Distribution
Random distribution refers to how individuals are placed into various positions entirely by chance. In this exercise, six individuals are seated around a table randomly, creating different potential message paths between A and B.
In circular setups, random distribution assures that each individual has equal chances of occupying any of the seats, ensuring that scenarios are unpredictably varied.
  • The concept of randomness simplifies the probability assessment, making analysis accessible by guaranteeing uniform probability across all configurations.
  • This uniformity is what allows the use of simplified calculations in predicting outcomes, like calculating expected values with reasonable certainty.
  • Random distribution is crucial in ensuring that our model remains fair and each permutation is equally probable.
Hence, appreciating the role of random distribution provides clarity on the unpredictability yet regularity maintained in circular permutation scenarios.

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Most popular questions from this chapter

Annie and Alvie have agreed to meet between 5:00 \(\mathrm{p} . \mathrm{M}\). and 6:00 P.M. for dinner at a local health-food restaurant. Let \(X=\) Annie's arrival time and \(Y=\) Alvie's arrival time. Suppose \(X\) and \(Y\) are independent with each uniformly distributed on the interval \([5,6]\). a. What is the joint pdf of \(X\) and \(Y\) ? b. What is the probability that they both arrive between \(5: 15\) and \(5: 45\) ? c. If the first one to arrive will wait only \(10 \mathrm{~min}\) before leaving to eat elsewhere, what is the probability that they have dinner at the health- food restaurant? [Hint: The event of interest is \(A=\left\\{(x, y):|x-y| \leq \frac{1}{6}\right\\}\).]

Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit \(A=8 \mathrm{oz}\) and upper limit \(B=10 \mathrm{oz}\). Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes \(n=5,10,20\), and 30 .

Let \(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}\), and \(X_{6}\) denote the numbers of blue, brown, green, orange, red, and yellow \(\mathrm{M} \& \mathrm{M}\) candies, respectively, in a sample of size \(n\). Then these \(X_{l}\) 's have a multinomial distribution. According to the M\&M Web site, the color proportions are \(p_{1}=.24, p_{2}=.13, p_{3}=.16\), \(p_{4}=.20, p_{5}=.13\), and \(p_{6}=.14\). a. If \(n=12\), what is the probability that there are exactly two M\&Ms of each color? b. For \(n=20\), what is the probability that there are at most five orange candies? [Hint: Think of an orange candy as a success and any other color as a failure.] c. In a sample of \(20 \mathrm{M} \& \mathrm{Ms}\), what is the probability that the number of candies that are blue, green, or orange is at least 10 ?

The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 27718 -year-old American males, the sample mean waist circumference was \(86.3 \mathrm{~cm}\). A somewhat complicated method was used to estimate various population percentiles, resulting in the following values: \(\begin{array}{lllllll}5^{\text {th }} & 10^{\text {th }} & 25^{\text {th }} & 50^{\text {th }} & 75^{\text {th }} & 90^{\text {th }} & 95^{\text {th }} \\\ 69.6 & 70.9 & 75.2 & 81.3 & 95.4 & 107.1 & 116.4\end{array}\) a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution. b. Suppose that the population mean waist size is \(85 \mathrm{~cm}\) and that the population standard deviation is \(15 \mathrm{~cm}\). How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least \(86.3 \mathrm{~cm}\) ? c. Referring back to (b), suppose now that the population mean waist size in \(82 \mathrm{~cm}\). Now what is the (approximate) probability that the sample mean will be at least \(86.3 \mathrm{~cm}\) ? In light of this calculation, do you think that \(82 \mathrm{~cm}\) is a reasonable value for \(\mu\) ?

A binary communication channel transmits a sequence of "bits" ( 0 s and 1s). Suppose that for any particular bit transmitted, there is a \(10 \%\) chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0 ). Assume that bit errors occur independently of one another. a. Consider transmitting 1000 bits. What is the approximate probability that at most 125 transmission errors occur? b. Suppose the same 1000 -bit message is sent two different times independently of one another. What is the approximate probability that the number of errors in the first transmission is within 50 of the number of errors in the second?
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