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Chapter 3: Problem 8

Each time a component is tested, the trial is a success ( \(S\) ) or failure \((F)\). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let \(Y\) denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of \(Y\), and state which \(Y\) value is associated with each one.

Short Answer

Expert verified
Outcomes: SSS (3), FSSS (4), SFFSSS/FSSSS (5), SSSFFS/FSSSF/FFSFSS (6), FFSSSS/FFSFS (7).

Step by step solution

01

Understand the Problem

We need to find the list of outcomes that correspond to the smallest possible values of trials, denoted as \( Y \), necessary to obtain three consecutive successes \( (S) \). The task is to identify the sequences of trials that achieve this.
02

Identify the Smallest Sequence for \( Y = 3 \)

If the first three trials are successes, the sequence is SSS. This is the smallest possible value, where \( Y = 3 \).
03

Identify the Sequence for \( Y = 4 \)

To need a fourth trial, the first three should not all be consecutive successes. The sequence can be FSSS, where a failure occurs first, so \( Y = 4 \).
04

Identify the Sequences for \( Y = 5 \)

To get \( Y = 5 \), we need either one initial success followed by a failure and then three successes, SFFSSS, or two initial failures followed by three successes, FSSSS. Both result in \( Y = 5 \).
05

Identify the Sequences for \( Y = 6 \)

Here, we can have multiple patterns like SSSFFS, FSSSF, or FFSFSS. They show how non-consecutive successes force more trials, leading to \( Y = 6 \).
06

Identify the Sequence for \( Y = 7 \)

The sequence FFSSSS or similar forms like FFSFS are needed to reach \( Y = 7 \), where initial trials include multiple failures before reaching three consecutive successes.

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Key Concepts

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

Component Testing
Component testing is a crucial aspect of understanding how individual parts or elements perform under specific conditions. In this context, we are observing whether each test results in a success (S) or a failure (F). Each test is like a mini-experiment where the component is evaluated based on its functionality. This helps us evaluate:
  • The reliability of the component
  • What conditions lead to performance success or failure
  • The likelihood of achieving three consecutive successes
Understanding component testing involves analyzing these results to predict outcomes and improve performance.
Sequential Trials
The concept of sequential trials refers to conducting tests one after another, following a specific order or pattern. Each trial can show varying results, and only the overall sequence can provide meaningful insights.
  • Each trial result depends on the real-time performance of the component.
  • Sequential trials allow us to monitor the sequence of outcomes to see if patterns emerge.
For example, in an exercise exploring the smallest value of trials needed to achieve three consecutive successes: we must observe the entire series of trials until we reach that criterion. Sequential trials ensure every test is recorded till the required outcome appears.
Consecutive Successes
Consecutive successes occur when success (S) is achieved in a row, enhancing the reliability of the trials. For this problem, we focus on getting three successive successes.
  • The value of each outcome sequence is determined by how quickly consecutive successes appear.
  • The goal is to logically deduce how many tests (Y) it takes to see this pattern.
The shortest potential sequence is straightforward: SSS, needing only three trials. Extending trials results in different paths to consecutive successes, demonstrating how this pattern can inform conclusions.
Outcome Sequences
An outcome sequence refers to the specific order in which successes and failures occur. These sequences help determine the efficiency and reliability of reaching desired outcomes.
  • Outcome sequences vary based on the arrangement and combination of successes and failures.
  • They are integral to calculating the minimum number of trials necessary to achieve the desired sequence.
In exercises looking to achieve a specific pattern of successes, like the three consecutive ones, knowing various outcome sequences is essential. They showcase different paths you could take to achieve similar results.

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Most popular questions from this chapter

The \(n\) candidates for a job have been ranked \(1,2,3, \ldots, n\). Let \(X=\) the rank of a randomly selected candidate, so that \(X\) has pmf $$ p(x)= \begin{cases}1 / n & x=1,2,3, \ldots, n \\ 0 & \text { otherwise }\end{cases} $$ (this is called the discrete uniform distribution). Compute \(E(X)\) and \(V(X)\) using the shortcut formula. [Hint: The sum of the first \(n\) positive integers is \(n(n+1) / 2\), whereas the sum of their squares is \(n(n+1)(2 n+1) / 6 .]\)

Write a general rule for \(E(X-c)\) where \(c\) is a constant. What happens when \(c=\mu\), the expected value of \(X\) ?

A chemical supply company currently has in stock \(100 \mathrm{lb}\) of a certain chemical, which it sells to customers in 5 -lb batches. Let \(X=\) the number of batches ordered by a randomly chosen customer, and suppose that \(X\) has pmf \begin{tabular}{l|llll} \(x\) & 1 & 2 & 3 & 4 \\ \hline\(p(x)\) & \(.2\) & \(.4\) & \(.3\) & \(.1\) \end{tabular} Compute \(E(X)\) and \(V(X)\). Then compute the expected number of pounds left after the next customer's order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of \(X_{\text {.] }}\)

A family decides to have children until it has three children of the same gender. Assuming \(P(B)=P(G)=.5\), what is the pmf of \(X=\) the number of children in the family?

Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, highpitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let \(X\) be the number among the first 6 examined that have a defective compressor. a. Calculate \(P(X=4)\) and \(P(X \leq 4)\) b. Determine the probability that \(X\) exceeds its mean value by more than 1 standard deviation. c. Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If \(X\) is the number among 15 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) \(P(X \leq 5)\) than to use the hypergeometric pmf.
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