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Chapter 12: Problem 37

Car Colors. See Exercise 1.25. Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for vehicles sold globally in 2018:16. \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Color & White & Black & Gray & Silver & Natural & Ried & Blue \\ \hline Probability & \(0.39\) & \(0.17\) & \(0.12\) & \(0.10\) & \(0.07\) & \(0.07\) & \(0.07\) \\ \hline \end{tabular} a. What is the probability that the vehicle you chose has any color other than those listed? b. What is the probability that a randomly chosen vehicle is neither white nor silver?

Short Answer

Expert verified
a. 0.01 b. 0.51

Step by step solution

01

Understanding the Problem

We are given the probabilities of selecting a new car of certain colors. We need to calculate the probability of choosing a car with colors not listed and the probability of a randomly selected car being neither white nor silver.
02

Calculate Total Probability of Listed Colors

Add the probabilities of all the listed colors. That is, the probability of a car being white, black, gray, silver, natural, red, or blue.\[P(\text{listed colors}) = P(\text{White}) + P(\text{Black}) + P(\text{Gray}) + P(\text{Silver}) + P(\text{Natural}) + P(\text{Red}) + P(\text{Blue}) = 0.39 + 0.17 + 0.12 + 0.10 + 0.07 + 0.07 + 0.07 = 0.99\]
03

Calculate Probability of Other Colors

To find the probability of a vehicle being a color other than those listed, subtract the total probability of listed colors from 1.\[P(\text{other colors}) = 1 - P(\text{listed colors}) = 1 - 0.99 = 0.01\]
04

Calculate Probability of Neither White Nor Silver

First, calculate the probability of a car being either white or silver by adding their probabilities together.\[P(\text{White or Silver}) = P(\text{White}) + P(\text{Silver}) = 0.39 + 0.10 = 0.49\]Then, subtract this from 1 to find the probability of the car being neither white nor silver.\[P(\text{Neither White nor Silver}) = 1 - P(\text{White or Silver}) = 1 - 0.49 = 0.51\]

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Key Concepts

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

Random Selection
Random selection is a fundamental concept in probability that refers to choosing an item from a set, where each item has an equal chance of being selected. In our example, we're randomly selecting a car or truck and noting its color. By doing this, we assume that each car color has a preset probability of being chosen, and these probabilities sum up to one.
Consider it like picking a sweet from a jar without looking, where each color (or flavor) is predetermined by probabilities. This randomness means you're just as likely to select a white car as you are any other color following its respective probability.
Color Distribution
Color distribution in the context of probability involves understanding the likelihood of different outcomes, in this case, the colors of sold vehicles. Each color has a probability assigned to it, which reflects how common or popular that color is in the population of new cars.
In our car color example, the distribution shows white has the highest probability at 0.39, indicating it's the most popular car color. This helps us quantify exactly how common each car color is on the road.
  • White: 0.39
  • Black: 0.17
  • Gray: 0.12
  • Silver: 0.10
  • Natural: 0.07
  • Red: 0.07
  • Blue: 0.07
Understanding this distribution is key to answering more complex probability questions.
Complement Rule
The complement rule is essential in probability. It helps us find the probability of an event not happening, by using the fact that the total probability of all possible outcomes must equal 1.
For example, in determining the probability of a vehicle being a color other than those listed, we calculated the complement. This is done by subtracting the total probability of all listed colors from 1. If the sum of listed colors is 0.99, the probability of a car being of a different color is 0.01, or \( P(\text{other colors}) = 1 - 0.99 = 0.01 \).
This rule is handy for quickly finding the probability of negations without complex calculations.
Probability Addition Rule
The probability addition rule is used when you're finding the probability of either one event or another happening. If the events are mutually exclusive (cannot occur at the same time), you simply add their probabilities.
In our example, to find the probability that a car is neither white nor silver, we first added the probabilities of white and silver to get 0.49 (\( P(\text{White}) + P(\text{Silver}) \)). Then, using the complement rule, we subtract from 1 to find that \( P(\text{Neither White nor Silver}) = 1 - 0.49 = 0.51 \). This rule allows quick computation of exclusion by calculating for inclusion first.

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Most popular questions from this chapter

A Flush. You read online that the probability of being dealt a flush (all five cards of the same suit) in a five-card poker hand is \(1 / 508\). Explain carefully what this means. In particular, explain why it does not mean that if you are dealt 508 five-card poker hands, one will be a flush.

Who Takes the GMAT? In many settings, the "rules of probability" are just basic facts about percentages. The Graduate Management Admission Test (GMAT) website provides the following information about the geographic region of citizenship of those who took the test in 2018: \(1.9 \%\) were from Africa; \(0.3 \%\) were from Australia and the Pacific Islands; \(2.4 \%\) were from Canada; \(14.3 \%\) were from Central and South Asia; \(36.1 \%\) were from East and Southeast Asia; \(1.7 \%\) were from Eastern Europe; \(3.2 \%\) were from Mexico, the Caribbean, and Latin America; \(2.2 \%\) were from the Middle East; \(30.3 \%\) were from the United States; and \(7.6 \%\) were from Western Europe. 4 a. What percentage of those who took the test in 2018 were from the Americas (either Canada, the United States, Mexico, the Caribbean, or Latin America)? Which rule of probability did you use to find the answer? b. What percentage of those who took the test in 2018 were from some other region than the United States? Which rule of probability did you use to find the answer?

Spelling Errors. Spell-checking software catches "nonword errors" that result in a string of letters that is not a word, as when "the" is typed as "teh." When undergraduates are asked to type a 250word essay (without spell- checking), the number \(X\) of nonword errors has the following distribution: \begin{tabular}{|l|c|c|c|c|c|} \hline Value of \(\boldsymbol{X}\) & 0 & 1 & 2 & 3 & 4 \\ \hline Probability & \(0.1\) & \(0.2\) & \(0.3\) & \(0.3\) & \(0.1\) \\ \hline \end{tabular} a. Is the random variable \(X\) discrete or continuous? Why? b. Write the event "at least one nonword error" in terms of \(X\). What is the probability of this event? c. Describe the event \(X \leq 2\) in words. What is its probability? What is the probability that \(X<2\) ?

Are They Disjoint? Which of the following pairs of events, \(A\) and \(B\), are disjoint? Explain your answers. a. A person is selected at random. \(A\) is the event "the person selected is less than age 18 "; \(B\) is the event "the person selected is age 18 or over." b. A person is selected at random. \(A\) is the event "the person selected earns more than \(\$ 100,000\) per year"; \(B\) is the event "the person selected earns more than \(\$ 250,000\) per year." c. A pair of dice are tossed. \(A\) is the event "one of the dice is a 3 "; \(B\) is the event "the sum of the two dice is 3 ."

You read in a book on poker that the probability of being dealt a straight in a five-card poker hand is \(1 / 255\). This means that a. if you deal millions of poker hands, the fraction of them that contain a straight will be very close to \(1 / 255\). b. if you deal 255 poker hands, exactly one of them will contain a straight. c. if you deal 25,500 poker hands, exactly 100 of them will contain a straight.
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