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Chapter 12: Problem 14

How Many Cups of Coffee? Choose an adult age 18 or over in the United States at random and ask, "How many cups of coffee do you drink on average per day?" Call the response \(X\) for short. Based on a large sample survey, here is a probability model for the answer you will get: 8 \begin{tabular}{|c|c|c|c|c|c|} \hline Number & 0 & 1 & 2 & 3 & 4 or more \\ \hline Probability & \(0.36\) & \(0.26\) & \(0.19\) & \(0.08\) & \(0.11\) \\ \hline \end{tabular} a. Verify that this is a valid finite probability model. b. Describe the event \(X<4\) in words. What is \(P(X<4)\) ? c. Express the event "have at least one cup of coffee on an average day" in terms of \(X\). What is the probability of this event?

Short Answer

Expert verified
a. The model is valid. b. \(X<4\) means less than 4 cups, \(P(X<4) = 0.89\). c. "At least one cup" is \(X \geq 1\), probability is 0.64.

Step by step solution

01

Verify Finite Probability Model

To establish that the given model is valid, check if the sum of all probabilities equals 1. The provided probabilities are:\[P(X=0) = 0.36, \quad P(X=1) = 0.26, \quad P(X=2) = 0.19, \quad P(X=3) = 0.08, \quad P(X \geq 4) = 0.11\] Calculate the total:\[0.36 + 0.26 + 0.19 + 0.08 + 0.11 = 1.00\] Since the total is 1, it confirms that the probability model is valid.
02

Describe the Event X

The event \(X<4\) refers to an individual drinking fewer than 4 cups of coffee per day. This means considering the probabilities when \(X = 0, 1, 2,\) or \(3\). Calculate \(P(X<4)\) by summing these probabilities:\[P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)\]\[P(X<4) = 0.36 + 0.26 + 0.19 + 0.08 = 0.89\]
03

Express and Calculate Event X鈮�1

The event "at least one cup of coffee on an average day" corresponds to \(X \geq 1\). To find this probability, sum the probabilities for \(X = 1, 2, 3,\) and \(4 \text{ or more}\):\[P(X \geq 1) = P(X=1) + P(X=2) + P(X=3) + P(X \geq 4)\]\[P(X \geq 1) = 0.26 + 0.19 + 0.08 + 0.11 = 0.64\]

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Key Concepts

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

Finite Probability
A finite probability model involves a limited, countable set of possible outcomes, each assigned a probability. These probabilities must meet certain criteria to form a valid model.
  • Firstly, each probability must be a number between 0 and 1, where 0 indicates an impossible event and 1 signifies certainty.
  • Secondly, the sum of all the probabilities must equal 1.
In our coffee-drinking example, we see the potential outcomes regarding cups of coffee per day: 0 cups, 1 cup, 2 cups, 3 cups, and 4 or more cups. Each has a designated probability, and when summed鈥�0.36, 0.26, 0.19, 0.08, and 0.11 respectively鈥攖he total is 1, confirming it's a valid finite probability model.
Event Description
When working with probability models, understanding how to describe and interpret events is key. An event is any collection of possible outcomes. For instance, the event \(X<4\) refers to survey responses where the number of cups of coffee consumed is less than 4.
  • This includes responses of 0, 1, 2, or 3 cups per day.
  • Each of these outcomes has its associated probability.
Describing an event precisely in words is crucial for defining what specific probabilities you are considering. For example, "less than 4 cups" helps you focus on the outcomes you need to calculate for specific probabilities.
Probability Calculation
Calculating the probability of an event involves summing up the probabilities of all outcomes that form part of the event.
For our event \(X<4\), the individual probabilities for 0, 1, 2, and 3 cups are added together:
\[ P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) \] \[ P(X<4) = 0.36 + 0.26 + 0.19 + 0.08 = 0.89 \] This tells us that there's an 89% chance a randomly chosen adult drinks fewer than 4 cups of coffee. Similarly, for the event "at least one cup of coffee", we sum the probabilities from 1 cup upwards:\[ P(X \geq 1) = 0.26 + 0.19 + 0.08 + 0.11 = 0.64 \] Thus, 64% of adults drink at least one cup daily.
Random Variable
In probability and statistics, a random variable represents potential outcomes of a random process. It is typically denoted by a capital letter, like \(X\) in our example. Here, \(X\) represents how many cups of coffee an adult drinks on average per day.

The random variable can take on multiple values, each corresponding to a particular probability:
  • 0 cups: 0.36 probability
  • 1 cup: 0.26 probability
  • 2 cups: 0.19 probability
  • 3 cups: 0.08 probability
  • 4 or more cups: 0.11 probability
Understanding what a random variable signifies and how it quantifies risk or uncertainty is essential. It encapsulates the idea of describing uncertainty in statistical terms, allowing us to analyze and interpret real-world phenomena comprehensively.

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Most popular questions from this chapter

Foreign-language Study. Choose a student in a U.S. public high school at random and ask if he or she is studying a language other than English. Here is the distribution of results: \begin{tabular}{|l|c|c|c|c|c|} \hline Language & Sparish & French & German & All athers & None \\ \hline Probability & \(0.30\) & \(0.08\) & \(0.02\) & \(0.03\) & \(0.57\) \\ \hline \end{tabular} a. Explain why this is a legitimate probability model. b. What is the probability that a randomly chosen student is studying a language other than English? c. What is the probability that a randomly chosen student is studying French, German, or Spanish?

Education Among Young A dults. Choose a young adult (aged 25-29) at random. The probability is \(0.07\) that the person chosen did not complete high school, \(0.46\) that the person has a high school diploma but no further education, and \(0.37\) that the person has at least a bachelor's degree. a. What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor's degree? b. What is the probability that a randomly chosen young adult has at least a high school education?

Overweight? Although the rules of probability are just basic facts about percentages or proportions, we need to be able to use the language of events and their probabilities. Choose an American adult aged 20 years or over at random. Define two events: \(A=\) the person chosen is obese \(B=\) the person chosen is overweight but not obese According to the National Center for Health Statistics, \(P(A)=0.40\) and \(P(B)=0.32\). a. Explain why events \(A\) and \(B\) are disjoint. b. Say in plain language what the event " \(A\) or \(B\) " is. What is \(P(A\) or \(B)\) ? c. If \(C\) is the event that the person chosen has normal weight or less, what is \(P(C)\) ?

Grades in a Business Course. Indiana University posts the grade distributions for its courses online. 11 Students in Business 100 in the spring 2019 semester received these grades: \(9 \%\) \(\mathrm{A}+, 15 \% \mathrm{~A}, 13 \% \mathrm{~A}-, 10 \% \mathrm{~B}+, 13 \% \mathrm{~B}, 8 \% \mathrm{~B}-7 \% \mathrm{C}+, 11 \% \mathrm{C}, 0 \% \mathrm{C}-, 2 \% \mathrm{D}+, 4 \% \mathrm{D}, 0 \% \mathrm{D}=\), and \(8 \%\) F. Choose a Business 100 student at random. To "choose at random" means to give every student the same chance to be chosen. The student's grade on a four-point scale (with A \(+=4.3\), \(\mathrm{A}=4, \mathrm{~A}-=3.7, \mathrm{~B}+=3.3, \mathrm{~B}=3.0, \mathrm{~B}-=2.7, \mathrm{C}+=2.3, \mathrm{C}=2.0, \mathrm{C}-=1.7, \mathrm{D}+=1.3, \mathrm{D}=1.0\), \(\mathrm{D}=0.7\), and \(\mathrm{F}=0.0\) ) is a random variable \(X\) with this probability distribution: \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Value of \(X\) & \(0.0\) & \(0.7\) & \(1.0\) & \(1.3\) & \(1.7\) & \(2.0\) & \(2.3\) & \(2.7\) & \(3.0\) & \(3.3\) & \(3.7\) & \(4.0\) & \(4.3\) \\ \hline Probability & \(0.08\) & \(0.00\) & \(0.04\) & \(0.02\) & \(0.00\) & \(0.11\) & \(0.07\) & \(0.08\) & \(0.13\) & \(0.10\) & \(0.13\) & \(0.15\) & \(0.09\) \\ \hline \end{tabular} a. Is \(X\) a finite or continuous random variable? Explain your answer. b. Say in words what the meaning of \(P(X \geq 3.0)\) is. What is this probability? c. Write the event "the st udent got a grade poorer than \(\mathrm{B}\)-" in terms of values of the random variable \(X\). What is the probability of this event?

Probability Says ... Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.) \(\begin{array}{lllllll}0 & 0.05 & 0.45 & 0.50 & 0.55 & 0.95 \quad 1\end{array}\) a. This event is impossible. It can never occur. b. This event is just as likely to occur as it is to not occur. c. This event is very likely, but it will not occur once in a while in a long sequence of trials. d. This event will occur slightly less often than not.
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