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Chapter 5: Problem 24

Six individuals, including \(\mathrm{A}\) and \(\mathrm{B}\), 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 \(\mathrm{A}\) and \(\mathrm{B}\) ) would you expect to handle the message?

Short Answer

Expert verified
The expected number of individuals to handle the message is 2.8.

Step by step solution

01

Determine Possible Arrangements

Since the six individuals (including A and B) are seated randomly around a circular table, A's seat number X and B's seat number Y can each be any integer from 1 to 6. However, for any arrangement, A can be assigned to seat 1 (due to the circular nature) without loss of generality, and Y would then take remaining values.
02

Calculate Distances between Seats

The shortest paths for the message to go from seat X to seat Y around a circle can be defined by calculating the distances in both clockwise and counterclockwise directions. Distance in clockwise direction for a pair (j, k) is calculated as \((k-j) \mod 6\), and counterclockwise as \((j-k) \mod 6\).
03

Establish Close Distance Condition

The message will be sent through the shortest path. For direction from X to Y, pick the minimum of the two calculated distances, \( \min((k-j) \mod 6, (j-k) \mod 6) \). Use modular arithmetic to normalize these values because of the circular arrangement.
04

Calculate Expected Value

The task now is to compute the expected number of individuals handling the message. This means calculating the expectation of the minimum distance value plus 1 (to include A in the count of handlers). Each configuration (X=1, Y=2 to 6) is considered, given the circular symmetry:- The distribution considers all five possibilities of B's seats and calculates the minimum distance between them. This can be analyzed stepwise:1. \(d = 1\): (Y=2 or Y=6), handled by 2 individuals.2. \(d = 2\): (Y=3 or Y=5), handled by 3 individuals.3. \(d = 3\): (Y=4), handled by 4 individuals.Calculate the weighted average of these scenarios to derive the expectation.
05

Solve for Expected Value Result

By considering possible distances, we plug in these into the expected value formula: \[E(Z) = \frac{2 \times 2 + 2 \times 3 + 1 \times 4}{5} = \frac{2 + 2 + 1}{5}\] Resulting in an expected value: \[E(Z) = \frac{14}{5} = 2.8\]

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Key Concepts

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

Modular Arithmetic
When dealing with a circular arrangement, like seats at a table, modular arithmetic becomes a powerful tool. Imagine numbers forming a circle rather than a straight line. If you move past the largest number, you circle back to the start.
This is what makes modular arithmetic perfect for problems like our circular table.

Let's break down some basics:
  • The expression \(a \mod b\) gives the remainder after dividing \(a\) by \(b\). For example, \(7 \mod 6 = 1\), because 7 divided by 6 leaves a remainder of 1.
  • In the context of our circular table problem, the smallest distance between any two seats, say seat \((j)\) and seat \((k)\), can be effectively computed using modular arithmetic.
  • Considering clockwise movement, the distance would be \((k-j) \mod 6\) and counterclockwise as \((j-k) \mod 6\).
  • The problem of deciding which direction is shorter can be solved by comparing these two modular results.
Understanding the flow of modular arithmetic allows us to handle circular permutations, which are crucial for determining the shortest message path.
Expectation Calculation
Expectation calculation in probability theory helps us figure out the average outcome over many trials. It's like finding a typical pattern or behavior in a random process.

In our circular table problem, we're interested in the average number of people expected to handle a message sent from person A to person B.
  • To determine this, we need to calculate the expected value: the average over all possible scenarios.
  • The expectation formula considers each possible distance the message might travel, weighted by how likely that scenario is to occur.
  • Specifically, we calculated the weighted sum of the number of message handlers for each possible seat position B could occupy (considering the shortest path).
By breaking down the possibilities (distances of 1, 2, or 3), and assigning the appropriate number of handlers to each, we arrive at the expected average. This concept is essential in probabilistic situations, allowing predictions about random events.
Circular Table Problem
The circular table problem we're tackling involves six people randomly seated around a round table, emphasizing relational positions rather than fixed points.

A few key points to understand:
  • The circular nature means we can "fix" one person's position, simplifying calculations without losing generality (i.e., assuming A is always in seat 1).
  • This setup leads us to consider paths between seats based on closeness, leveraging modular arithmetic for distance calculation.
  • Practically, for person A to pass a message to person B, we calculate both clockwise and counterclockwise paths to find the shortest, effectively determining how many people (including A and B) will handle the message along this path.
  • Because of the circular symmetry, the problem's results remain consistent no matter where you begin, reducing the problem's complexity while maintaining accuracy.
By dissecting this problem, students learn to apply mathematical concepts to real-world scenarios, gaining insights into symmetry and optimization strategies. This aids not only in solving puzzles like this but also in developing problem-solving frameworks useful in complex organizational tasks.

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, \ldots, X_{100}\) denote the actual net weights of 100 randomly selected 50 -lb bags of fertilizer. a. If the expected weight of each bag is 50 and the variance is 1 , calculate \(P(49.9 \leq \bar{X} \leq 50.1\) ) (approximately) using the CLT. b. If the expected weight is \(49.8 \mathrm{lb}\) rather than \(50 \mathrm{lb}\) so that on average bags are underfilled, calculate \(P(49.9 \leq \bar{X} \leq 50.1)\).

A box contains ten sealed envelopes numbered \(1, \ldots, 10\). The first five contain no money, the next three each contains \(\$ 5\), and there is a \(\$ 10\) bill in each of the last two. A sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \(X_{1}, X_{2}\), and \(X_{3}\) denote the amounts in the selected envelopes, the statistic of interest is \(M=\) the maximum of \(X_{1}, X_{2}\), and \(X_{3}\). a. Obtain the probability distribution of this statistic. b. Describe how you would carry out a simulation experiment to compare the distributions of \(M\) for various sample sizes. How would you guess the distribution would change as \(n\) increases?

Annie and Alvie have agreed to meet for lunch between noon (0:00 P.M.) and 1:00 P.M. Denote Annie's arrival time by \(X\), Alvie's by \(Y\), and suppose \(X\) and \(Y\) are independent with pdf's $$ \begin{aligned} &f_{X}(x)=\left\\{\begin{array}{cl} 3 x^{2} & 0 \leq x \leq 1 \\ 0 & \text { otherwise } \end{array}\right. \\ &f_{Y}(y)=\left\\{\begin{array}{cl} 2 y & 0 \leq y \leq 1 \\ 0 & \text { otherwise } \end{array}\right. \end{aligned} $$ What is the expected amount of time that the one who arrives first must wait for the other person? [Hint: \(h(X, Y)=\) \(|X-Y| .]\)

A particular brand of dishwasher soap is sold in three sizes: \(25 \mathrm{oz}, 40 \mathrm{oz}\), and \(65 \mathrm{oz}\). Twenty percent of all purchasers select a 25 -oz box, \(50 \%\) select a 40 -oz box, and the remaining \(30 \%\) choose a \(65-\mathrm{oz}\) box. Let \(X_{1}\) and \(X_{2}\) denote the package sizes selected by two independently selected purchasers. a. Determine the sampling distribution of \(\bar{X}\), calculate \(E(\bar{X})\), and compare to \(\mu\). b. Determine the sampling distribution of the sample variance \(S^{2}\), calculate \(E\left(S^{2}\right)\), and compare to \(\sigma^{2}\).

A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1,10 by supplier 2 , and 12 by supplier 3. Six of these are to be randomly selected for a particular assembly. Let \(X=\) the number of supplier 1s components selected, \(Y=\) the number of supplier \(2 \mathrm{~s}\) components selected, and \(p(x, y)\) denote the joint pmf of \(X\) and \(Y\). a. What is \(p(3,2)\) ? [Hint: Each sample of size 6 is equally likely to be selected. Therefore, \(p(3,2)=\) (number of outcomes with \(X=3\) and \(Y=2) /\) (total number of outcomes). Now use the product rule for counting to obtain the numerator and denominator.] b. Using the logic of part (a), obtain \(p(x, y)\). (This can be thought of as a multivariate hypergeometric distributionsampling without replacement from a finite population consisting of more than two categories.)
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