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Chapter 4: Problem 33

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution. \(\bullet\) Let \(X=\) the number of years a student will study ballet with the teacher. \(\bullet\) Let \(P(x)=\) the probability that a student will study ballet \(x\) years. What does the column \(" P(x)\) "sum to and why?

Short Answer

Expert verified
The column "P(x)" sums to 1 because it represents a probability distribution of all potential outcomes.

Step by step solution

01

Understanding Probability Distribution

In a probability distribution, each outcome has a probability assigned to it. For any discrete random variable, the sum of all the probabilities must equal 1. This property ensures that one of the possible outcomes must occur.
02

Analyzing the Problem

Here, the column "P(x)" represents the probabilities of the number of years a student will study ballet, denoted by the variable X. Since X is a random variable representing all possible outcomes, P(x) values represent a discrete probability distribution.
03

Applying Probability Distribution Property

Given that P(x) forms a probability distribution for the number of years students study ballet, the sum of all probabilities from P(x) must equal 1. This is because they represent the exhaustive set of all outcomes for how long a student studies ballet, ensuring that a student fits one of the possible year categories.

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Key Concepts

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

Understanding Discrete Random Variables
In statistics, a discrete random variable is a type of variable that has distinct, separate values. Unlike continuous variables, which can take on any value within a range, discrete random variables have specific, countable outcomes. For example, in a ballet class, the number of years a student can potentially study is a discrete random variable. Each year is a separate outcome that can be measured individually.
  • A discrete random variable, like the number of years a student studies ballet, can be denoted by \( X \).
  • Each possible number of years forms part of the variable's possible outcomes.
  • The probability of each individual outcome is defined by the probability distribution \( P(x) \).
Thus, in the context of a ballet instructor planning her classes, understanding that \( X \) is a discrete random variable helps in forecasting how many students might continue their studies for a given number of years.
Importance of Probability Sum
In probability theory, when dealing with a discrete random variable, the sum of all probabilities must equal 1. This rule ensures that all possible outcomes for a variable are considered, capturing the entire sample space's probability. For the ballet instructor assessing student continuation:
  • Each probability \( P(x) \) reflects the chance of students studying for \( x \) years.
  • The condition \( \sum P(x) = 1 \) confirms that these probabilities account for every possible study duration.
  • This is crucial for making accurate predictions about future class sizes and offerings.
In simpler terms, when you add up all the probabilities of every possible event (like each year a student might study), the total is always 1. This confirms that some outcome will certainly occur.
Ballet Class Planning with Probability
Effective class planning is vital for any instructor, including those teaching ballet. By using probability distributions, instructors like our ballet teacher can better understand how many students are likely to continue their studies each year. Imagine planning a ballet curriculum without knowing how many students will progress to the next level:
  • The probability distribution gives a clear idea of student retention rates.
  • It's essential to plan resources, schedule classes, and prepare materials in line with expected student numbers.
  • For example, if the probability of a student studying for three years is relatively high, the instructor might prepare more advanced classes, knowing there will be sufficient interest.
In sum, employing tools like discrete random variables and probability sums allows ballet instructors to strategically plan for the future, ensuring they can provide quality education tailored to their students' needs and needs.

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Most popular questions from this chapter

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let \(X =\) the number of people who are literate. a. Sketch a graph of the probability distribution of \(X\). b. Using the formulas, calculate the (i) mean and (ii) standard deviation of \(X.\) c. Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ____ (__,___) d. On average, how many dealerships would we expect her to have to call until she finds one that has the car? e. Find the probability that she must call at most four dealerships. f. Find the probability that she must call three or four dealerships.

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given as in Table 4.34. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 3 & {0.05} \\ \hline 4 & {0.40} \\ \hline 5 & {0.30} \\ \hline 6 & {0.15} \\ \hline 7 & {0.10} \\\ \hline\end{array}$$ On average, how many years do you expect it to take for an individual to earn a B.S.?

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of \(X . X \sim\) _____(___,___) d. On average, how many schools would you expect to offer such courses? e. Find the probability that at most ten offer such courses. f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status. In words, define the random variable \(X\).
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