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Chapter 3: Problem 2

An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 3 of these automobiles at random, list the elements of the sample space \(S\) using the letters \(B\) and \(N\) for blemished and nonblemished, respectively; then to each sample point assign a value \(x\) of the random variable \(X\) representing the number of automobiles purchased by the agency with paint blemishes.

Short Answer

Expert verified
The final sample space is \(S = \{BBN, BNB, NBB, BNN, NBN, NNB\}\), with the corresponding random variable 'X' values: \(X = \{2, 2, 2, 1, 1, 1\}\)

Step by step solution

01

Identify All Possible Outcomes

Consider each car individually. Select three cars out of five. Each car either has a blemish (B) or it doesn't (N). The different combinations of size 3 we can obtain are: BBB, BBN, BNN, NBB, NNB, NNN. In case of three cars being selected at once, the scenarios could be: BBB, BBN, BNN, NNB
02

Assign Values to the Random Variable 'X'

The random variable 'X' is defined as the number of automobiles with blemishes. Now assign a value to 'X' for each sample point. For BBB, X=3. For BBN and NBB, X=2. For BNN and NNB, X=1. For NNN, X=0. It's necessary to note that not all sample points may be possible considering there are only 2 blemished cars and 3 non-blemished cars in total. Specifically, the BBB and NNN outcomes are not possible.
03

Final Sample Space and Corresponding 'X' Values

Considering the available cars, the final sample space is: \(S = \{BBN, BNB, NBB, BNN, NBN, NNB\}\). The corresponding random variable 'X' values are \(X = \{2, 2, 2, 1, 1, 1\}\)

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Key Concepts

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

Sample Space
In probability and statistics, a **sample space** is the set of all possible outcomes of an experiment. When determining the sample space for a situation, we list each possible scenario.
In our case, the experiment involves selecting three automobiles from a group of five, where there are two blemished (B) and three non-blemished (N) cars.
The elements of the sample space are all the ways we can choose three cars, considering the blemishes:
  • BBN
  • BNB
  • NBB
  • BNN
  • NBN
  • NNB
Each combination shows one possible group of cars that might be chosen by the agency. Understanding the sample space helps us comprehend all potential outcomes and lays the foundation for calculating probabilities.
Random Variable
A **random variable** is a numerical value that describes the outcomes of a statistical experiment. In this context, our random variable 'X' represents the number of cars with paint blemishes in the selected group of three automobiles.
After listing the sample space, we assign a value to 'X' for each possible outcome:
  • BBN, BNB, NBB all result in X = 2, because two blemished cars are chosen.
  • BNN, NBN, NNB all result in X = 1, because only one blemished car is chosen.
The random variable helps us translate outcomes into quantifiable data, which is essential for calculating probabilities and making statistical inferences.
Combinatorics
**Combinatorics** is a branch of mathematics dealing with combinations and permutations of objects. It is crucial in determining the sample space by calculating the number of ways to select automobiles.
In the given exercise:
  • We calculate how we can select three cars from a total of five cars.
Applying combinatorial analysis simplifies complex selections and ensures that all possibilities are considered. This mathematical method is often used in probability to systematically analyze situations where multiple arrangements are possible.
Blemished Automobiles
The term **blemished automobiles** refers to cars that have slight imperfections in their paint. In the exercise, two out of five cars are blemished, which influences the composition of the sample space.
When selecting three cars, understanding the distribution of blemished and non-blemished cars is essential:
  • It informs the probabilities related to each outcome.
  • It impacts the value of the random variable, helping us assess the likelihood of receiving blemished cars.
Recognizing the role of blemished automobiles in probability calculations highlights the practical applications of statistics, helping to make informed decisions based on available information.

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Most popular questions from this chapter

Suppose a special type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The pdf that characterizes the proportion \(Y\) that make a profit is given by $$ f(x)=\left\\{\begin{array}{ll} k y^{4}(1-y)^{3}+ & 0 \leq y \leq 1, \\ 0_{1} & \text { elsewhere. } \end{array}\right. $$ (a) What is the value of \(k\) that renders the above a valid density function? (b) Find the probability that at most \(50 \%\) of the firms make a profit in the first year. (c) Find the probability that at least \(80 \%\) of the firms make a profit in the first year.

Consider the following joint probability density function of the random variables \(X\) and \(Y:\) $$ f\\{x, y)=\left\\{\begin{array}{ll} \frac{3 x-y}{9}, & 1 < x < 3,1< y<2, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find the marginal density functions of \(X\) and \(Y\). (b) Are \(X\) and \(Y\) independent? (c) Find \(P(X>2)\).

Find a formula for the probability distribution of the random variable \(X\) representing the outcome when a single die is rolled once.

A coin is tossed twice. Let \(Z\) denote the number of heads on the first toss and \(W\) the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a \(40 \%\) chance of occurring, find (a) the joint probability distribution of \(W\) and \(Z\); (b) the marginal distribution of \(W\); (c) the marginal distribution of \(Z\); (d) the probability that at least 1 head occurs.

Determine the value \(c\) so that each of the following functions can serve as a probability distribution of the discrete random variable \(X\) : (a) \(f(x)=c\left(x^{2}+4\right),\) for \(a:=0,1,2,3\) (b) \(f(x)=c\left(\begin{array}{c}2 \\\ x\end{array}\right)\left(\begin{array}{c}3 \\ 3-x\end{array}\right),\) for \(z=0,1,2\)
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