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Chapter 3: Problem 11

An automobile service facility specializing in engine tune-ups knows that \(45 \%\) of all tune-ups are done on fourcylinder automobiles, \(40 \%\) on six- cylinder automobiles, and \(15 \%\) on eight-cylinder automobiles. Let \(X=\) the number of cylinders on the next car to be tuned. a. What is the pmf of \(X\) ? b. Draw both a line graph and a probability histogram for the pmf of part (a). c. What is the probability that the next car tuned has at least six cylinders? More than six cylinders?

Short Answer

Expert verified
The pmf lists: \(P(X=4)=0.45\), \(P(X=6)=0.40\), \(P(X=8)=0.15\). Probability of at least six cylinders is 0.55; more than six is 0.15.

Step by step solution

01

Define the Random Variable and Probabilities

The random variable, \(X\), represents the number of cylinders on the next car to be tuned. The probabilities associated with each cylinder type are given as follows:- \(P(X=4) = 0.45\)- \(P(X=6) = 0.40\)- \(P(X=8) = 0.15\).
02

Probability Mass Function (pmf)

The pmf of a discrete random variable \(X\) lists the probabilities for each possible value of \(X\). Here, the pmf is:\[P(X=x) = \begin{cases} 0.45, & \text{if } x = 4 \0.40, & \text{if } x = 6 \0.15, & \text{if } x = 8 \0, & \text{otherwise}\end{cases}\]
03

Drawing the Line Graph

To draw a line graph for the pmf, plot the values of \(X\) (4, 6, 8) on the x-axis and their respective probabilities on the y-axis. Connect the points with lines.
04

Drawing the Probability Histogram

For a histogram, draw bars centered on each value of \(X\) (4, 6, 8) with heights equal to their probabilities (0.45, 0.40, 0.15). Make sure the widths are the same for each cylinder type.
05

Calculating Probability for At Least Six Cylinders

The probability that the next car tuned has at least six cylinders (\(X \geq 6\)) is:\[P(X \geq 6) = P(X = 6) + P(X = 8) = 0.40 + 0.15 = 0.55\].
06

Calculating Probability for More Than Six Cylinders

The probability that the next car tuned has more than six cylinders (\(X > 6\)) is:\[P(X > 6) = P(X = 8) = 0.15\].

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Key Concepts

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

Discrete Random Variable
In probability theory, a discrete random variable is a variable that can take a countable number of distinct values. For instance, the outcomes of rolling a six-sided die (1 to 6) are discrete, as each possible roll is a distinct value. Discrete random variables are fundamental in probability because they help define the probability mass function. In the context of this exercise, the random variable \(X\) represents the number of cylinders in a car being tuned.
  • \(X = 4\) cylinders with probability 0.45
  • \(X = 6\) cylinders with probability 0.40
  • \(X = 8\) cylinders with probability 0.15
Each value of \(X\) corresponds to the number of cylinders a car might have. The pmf gives us a way to see which outcomes are more or less likely.
Probability Histogram
A probability histogram is a visual representation of a probability mass function (pmf) for a discrete random variable. In the probability histogram:
  • The x-axis represents the different outcomes (or values of the random variable), such as 4, 6, and 8 cylinders in this exercise.
  • The y-axis represents the probabilities of each outcome.
  • Each bar's height corresponds to the probability of its respective outcome.
The width of the bars in a probability histogram should be uniform, with each bar centered over its corresponding value. For our exercise, the bar for 4 cylinders is the tallest, indicating the highest probability of 0.45. Following are shorter bars for 6 cylinders with a probability of 0.40, and the shortest bar for 8 cylinders with a 0.15 probability.
Cylinder Probabilities
Cylinder probabilities in this exercise refer to the probability distribution over different numbers of cylinders in vehicles. These probabilities give insights into how often each type is serviced at the facility. Understanding cylinder probabilities has practical implications:
  • It helps the service center prepare resources based on the most common car types.
  • It allows for better inventory management of parts specific to four, six, or eight-cylinder cars.
The probabilities, as shown in the pmf, inform us that a significant portion of cars tuned will be four-cylinder (45%), followed by six-cylinder (40%), and the least likely are eight-cylinders (15%). Knowing these probabilities enhances operational efficiency and customer service in the automotive facility.
Line Graph
A line graph is a straightforward way to visualize how probabilities change over different values of a discrete random variable. To create a line graph for a pmf:
  • Plot the values of the random variable on the x-axis. For our case, these values are 4, 6, and 8 cylinders.
  • Plot the corresponding probabilities on the y-axis.
  • Each value-probability pair will form a point on the graph.
  • Connect these points with straight lines to form the graph.
Unlike a probability histogram, a line graph connects these points to illustrate possible trends or the relative differences between values. This type of graph helps quickly identify which events are more or less likely. In our exercise, the line graph would show peaks at 4 and 6, indicating these outcomes have higher probabilities than the one at 8 cylinders.

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Most popular questions from this chapter

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