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Chapter 2: Problem 6

A message can follow different paths through servers on a network. The sender's message can go to one of five servers for the first step; each of them can send to five servers at the second step; each of those can send to four servers at the third step; and then the message goes to the recipient's server. a. How many paths are possible? b. If all paths are equally likely, what is the probability that a message passes through the first of four servers at the third step?

Short Answer

Expert verified
There are 100 possible paths and the probability is \( \frac{1}{4} \).

Step by step solution

01

Understanding the Path Choices

For the first step, the message can be sent to 5 servers. For the second step, whichever server is chosen in the first step, it can go to any of the 5 servers. For the third step, each of the servers chosen in the second step can send the message to any of 4 servers.
02

Calculating Total Number of Paths

The total number of paths is calculated by multiplying the number of choices at each step. This gives us: \[ 5 \times 5 \times 4 \] Calculating this, we get \[ 5 \times 5 = 25 \]Then multiply by 4,\[ 25 \times 4 = 100 \] Thus, there are 100 possible paths for the message.
03

Calculating Paths through a Specific Third Step Server

We need to calculate the number of paths that go through the first of the four servers at the third step. For every choice of the first and second steps, there is exactly one choice to go to this specific server at the third step. So the number of paths through the first server at the third step is:\[ 5 \times 5 \times 1 = 25 \]
04

Calculating the Probability

Now, we calculate the probability that a message passes through the first of the four servers at the third step. The probability is given by the number of favorable paths divided by the total number of paths. Thus the probability is:\[ \frac{25}{100} = \frac{1}{4} \]

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Key Concepts

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

Network Paths
Understanding network paths can significantly aid in navigating and optimizing data routes. In this context, a message travelling through network servers takes different paths. Each path is a sequence the message follows, dictated by the servers it passes through step-by-step.
  • First Step: The message can initially travel to one of five available servers.
  • Second Step: Regardless of the server it reached first, it can proceed to any of the five servers again.
  • Third Step: From the second server, the message can go to one of four possible servers before reaching the recipient's server.
The final path does not reuse servers from previous steps, simplifying the choices and ensuring efficient data transmission. Thinking of network paths in this multi-step way allows us to visualize and count the possible routes effectively.
Combinatorial Calculation
Combinatorial calculations help us determine the total possible combinations while navigating paths. For network path calculations, this involves multiplying the choices available at each step. Let's break it down:
- **First step:** 5 choices 鈫� \(5\)- **Second step:** Each option in the first step leads to 5 more choices 鈫� \(5 imes 5 = 25\)- **Third step:** Each second-step choice opens 4 final choices 鈫� \(25 imes 4 = 100\)
This approach is efficient and proves that there are 100 unique paths the message can follow in total, demonstrating the power of combinatorial calculations to solve such pathfinding problems in networks.
Server Route Probability
Probabilities offer a way to understand the likelihood of specific routes being followed within a network. When all paths have equal chances of being chosen, calculating the probability of a specific route is straightforward. To do this, apply the ratio of favorable paths to total paths.
- Suppose we want the probability of the message passing through the first server in the third step.
- For each combination of the first two steps, there's precisely one way to reach that specific server in the third step. Thus, there are \(25\) favorable paths \(5 \times 5 imes 1 = 25\).
- Total possible paths: \(100\)
- Probability of passing through the first server: \(\frac{25}{100} = \frac{1}{4}\)
Understanding this probability calculation helps in decision-making processes regarding data routing strategies and network design, guiding towards efficient load balancing and resource allocation.

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Most popular questions from this chapter

A test of a printed circuit board uses a random test pattern with an array of 10 bits and each is equally likely to be 0 Or 1. Assume the bits are independent. a. What is the probability that all bits are \(1 \mathrm{~s}\) ? b. What is the probability that all bits are 0 s? c. What is the probability that exactly 5 bits are \(1 \mathrm{~s}\) and 5 bits are 0 s?

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