/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 The following table shows the ty... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 35

The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009 , Vol. 7, pp. \(111-127\) ). $$ \begin{array}{lrc} \text { Geological } & \text { Number of } & \text { Nonfailed } \\ \text { Formation Group } & \text { Nonfailed Wells } & \text { Well Depth } \\\ \text { Gneiss } & 1,515 & 255 \\ \text { Granite } & 26 & 218 \\ \text { Loch Raven Schist } & 3,290 & 317 \\ \text { Mafic } & 349 & 231 \\ \text { Marble } & 280 & 267 \\ \text { Prettyboy Schist } & 1,343 & 255 \\ \text { Other schists } & 887 & 267 \\ \text { Serpentine } & 36 & 217 \\ \text { Total } & 7,726 & 2,027 \end{array} $$ Calculate the probability mass function of depth for nonfailed wells from the table.

Short Answer

Expert verified
Depth 255: 2,858/7,726; Depth 218: 26/7,726; Depth 317: 3,290/7,726; Depth 231: 349/7,726; Depth 267: 1,167/7,726; Depth 217: 36/7,726.

Step by step solution

01

Understand the Task

We need to calculate the probability mass function (PMF) for the depth of nonfailed wells. This implies finding the probability of each possible depth that a nonfailed well might have.
02

Determine Frequency of Each Depth

Identify how often each well depth appears in the table. From the table, the only recorded depths are 255, 218, 317, 231, 267, and 217, each corresponding to different formations.
03

Calculate Total Number of Nonfailed Wells

The total number of nonfailed wells, as given in the table, is 7,726. This is the denominator for our PMF calculation for each depth.
04

Determine PMF at Each Depth

Calculate the probability for each depth by dividing the number of wells with that specific depth by 7,726. For example, for depth 255: add wells from 'Gneiss' and 'Prettyboy Schist' (1,515 + 1,343 = 2,858), then divide by 7,726 to get the PMF. Repeat similar calculations for all other depths.
05

Compile PMF Results

List the depths and their respective probabilities:* Depth 255: Probability = \(\frac{2,858}{7,726}\)* Depth 218: Probability = \(\frac{26}{7,726}\)* Depth 317: Probability = \(\frac{3,290}{7,726}\)* Depth 231: Probability = \(\frac{349}{7,726}\)* Depth 267: Probability = \(\frac{280 + 887}{7,726}\)* Depth 217: Probability = \(\frac{36}{7,726}\)
06

Ensure Correctness

Verify if the sum of probabilities is approximately 1 as a check.

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Key Concepts

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

Nonfailed Wells
Nonfailed wells refer to the subset of wells that have been successfully drilled without any failures. Understanding their characteristics can provide useful insights for geologists and engineers alike. In this exercise, we focus on the depths of nonfailed wells across various geological formations in Baltimore County. The success of these wells is critical because they offer valuable data that can inform future drilling projects. In particular, knowing the typical depth for nonfailed wells in different geological formations helps estimate the success rate and reliability of drilling in similar conditions.
Geological Formations
Geological formations significantly influence the characteristics of wells, including their depth. Each formation type, such as 'Gneiss', 'Granite', or 'Loch Raven Schist', has distinct features that affect drilling outcomes.
Here are the geological formations mentioned in the exercise:
  • Gneiss: Known for its strong composition, found mainly in metamorphic rock areas.
  • Granite: A coarse-grained igneous rock, which is known to be stable and resilient.
  • Loch Raven Schist: Typically has a foliated texture, which can affect drilling patterns.
  • Mafic: These rocks are rich in magnesium and iron, featuring unique drilling characteristics.
  • Marble: A metamorphic rock with a crystalline structure, which impacts well stability.
  • Prettyboy Schist & Other Schists: Known for their layered structure, these formations are commonly encountered in geology.
  • Serpentine: Characterized by its greenish coloration, influencing how drilling is handled.
Each of these formations presents a unique set of challenges and opportunities for drilling nonfailed wells. Understanding the interaction between the geological formation and well characteristics is essential for accurate geological assessment and successful drilling projects.
Probability Calculation
Probability calculation for nonfailed wells entails determining the likelihood of each well depth being encountered. This process uses the probability mass function (PMF), which is a way of expressing probabilities for discrete outcomes, like well depths.
Here are the steps for calculating the PMF in this context:
  • Identify distinct well depths as per the data provided (e.g., 255, 218).
  • Sum the number of wells with each specific depth to determine the frequency.
  • Divide the frequency of each depth by the total number of nonfailed wells (7,726 in this case) to find their probabilities.
For example, to find the PMF for a depth of 255 feet, add up the number of wells from the formations that share this depth, and divide the result by 7,726. Repeating this process for each depth yields a complete set of probabilities, detailing how prevalent each depth is among nonfailed wells. The sum of all probabilities should equal 1, ensuring you've accounted for all outcomes. This step-by-step calculation not only helps in identifying the most common depths but also aids in strategic planning regarding where to drill next based on historical success rates.

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Most popular questions from this chapter

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