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Chapter 3: Problem 1

A concrete beam may fail either by shear \((S)\) or flexure \((F)\). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let \(X=\) the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of \(X\).

Short Answer

Expert verified
There are eight outcomes: (SSS), (SSF), (SFS), (SFF), (FSS), (FSF), (FFS), (FFF), with \(X\) values of 3, 2, 2, 1, 2, 1, 1, 0 respectively.

Step by step solution

01

Identify Possible Outcomes

Each beam can fail by either shear (S) or flexure (F). Given three beams, each can either have an S or an F outcome, hence there are a total of \(2^3 = 8\) possible outcomes for the set of three beams.
02

List the Sample Space

Write down all the permutations of outcomes for the three beams as ordered triples. The outcomes are (SSS), (SSF), (SFS), (SFF), (FSS), (FSF), (FFS), and (FFF).
03

Assign Values to X

For each outcome, count the number of 'S' (shear) outcomes and assign this count to the variable \(X\). - (SSS) has 3 shears, so \(X = 3\).- (SSF) has 2 shears, so \(X = 2\).- (SFS) has 2 shears, so \(X = 2\).- (SFF) has 1 shear, so \(X = 1\).- (FSS) has 2 shears, so \(X = 2\).- (FSF) has 1 shear, so \(X = 1\).- (FFS) has 1 shear, so \(X = 1\).- (FFF) has 0 shears, so \(X = 0\).
04

Summarize the Sample Space with X Values

The complete sample space along with the corresponding values of \(X\) is:- (SSS) corresponds to \(X = 3\)- (SSF) corresponds to \(X = 2\)- (SFS) corresponds to \(X = 2\)- (SFF) corresponds to \(X = 1\)- (FSS) corresponds to \(X = 2\)- (FSF) corresponds to \(X = 1\)- (FFS) corresponds to \(X = 1\)- (FFF) corresponds to \(X = 0\)

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Key Concepts

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

Understanding Sample Space
In probability, a **sample space** is the set of all possible outcomes of a random experiment. It encompasses every potential scenario that could arise from the conditions of the exercise.
For our concrete beam problem, each beam can fail in one of two ways: **shear (S)** or **flexure (F)**. With each beam having these two possibilities, the sample space for three beams is composed of all sequences of these outcomes.
Hence, as we calculated, there are 8 potential outcomes in this sample space since there are 2 options per beam and 3 beams in total, hence:
  • (SSS)
  • (SSF)
  • (SFS)
  • (SFF)
  • (FSS)
  • (FSF)
  • (FFS)
  • (FFF)
Each outcome here is an ordered triplet, signifying the type of failure for the three beams selected. Understanding sample space allows us to compute probabilities for different events based on these outcomes.
The Concept of Random Selection
**Random selection** in probability refers to choosing items from a set without any bias or predictable pattern, ensuring each member of a set has an equal chance of being selected.
The key principle here is impartiality in the selection process, leading to each outcome in the sample space having the same probability. In the context of our beam problem, randomly selecting three failed beams means each potential outcome (each combination of shear and flexure) is equally likely.
This randomness is crucial because it maintains the integrity of the probability calculations. If the selection were not random, certain outcomes might become more or less likely, skewing the results and misleading any analyses based on these selections.
Types of Failure in Beams
In our scenario, the beams can fail in two main **types of failure**: **shear (S)** and **flexure (F)**. - Shear failure occurs when the material fails due to force parallel to the surface, causing layers to slide against each other. This is a sideways failure that often happens abruptly.- Flexure failure, on the other hand, involves bending. This type of failure happens when the tensile stress in the beam exceeds its capacity to resist bending, leading to a gradual deformation until failure.Recognizing these types of failure is crucial when assessing the integrity of structures and the effectiveness of materials. By quantifying the occurrences of each type within selected samples, as we do with variable \(X\), we can better understand structural vulnerabilities and inform design improvements.

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Most popular questions from this chapter

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