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Chapter 4: Problem 8

Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {0.15} \\ \hline 2 & {0.35} \\ \hline 3 & {0.40} \\ \hline 4 & {0.10} \\ \hline\end{array}$$ What is the probability the baker will sell exactly one batch? \(P(x=1)=\)_________

Short Answer

Expert verified
The probability is 0.15.

Step by step solution

01

Understand the Problem

We are asked to find the probability that the baker will sell exactly one batch of muffins. The probability distribution table provides probabilities for different numbers of batches sold.
02

Locate the Probability

From the provided probability distribution, find the probability associated with selling exactly one batch of muffins. This is directly available from the table under the column for \( x=1 \).
03

Read the Probability

According to the table, when \( x=1 \), the probability \( P(x=1) \) is 0.15.
04

Conclude with the Probability

The probability the baker will sell exactly one batch of muffins is 0.15.

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Key Concepts

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

Probability
Probability is the likelihood of an event happening. It is a measure that tells us how often we can expect a particular outcome to occur, out of all possible outcomes. In this exercise, the baker is trying to predict how many batches of muffins he can expect to sell based on the probability of different batch quantities being sold. To put it simply, probability can range from 0 to 1:
  • A probability of 0 means the event will not happen.
  • A probability of 1 means the event is certain to happen.
In our muffin scenario, the baker has figured out these probabilities through observation. For example, he knows there's a probability of 0.15 that exactly one batch will be sold. Understanding probability helps the baker prepare the right amount of muffins, reducing waste and increasing sales.
Probability Table
A probability table is a handy tool that organizes data so we can easily see different outcomes and their associated probabilities. In this muffin example, the table summarizes various scenarios of batch sales and the likelihood of each. The table displays:
  • Different values of batches (e.g., 1, 2, 3, 4).
  • The probability associated with each of these values.
This setup helps the baker quickly identify the chances for each potential outcome. For example, if he wants to find the probability of selling exactly one batch, he can simply look for the corresponding value in the table. This visual layout simplifies understanding of how likely each scenario is.
Discrete Random Variable
A discrete random variable is a type of random variable that takes on a countable number of distinct values. Each value of this variable has a specific probability associated with it. In our baker's case, the number of muffin batches sold is a discrete random variable, as it can only be specific whole numbers like 1, 2, 3, and so on. For discrete random variables:
  • The values are distinct and countable.
  • Each value has an associated probability.
  • The probabilities add up to 1, representing all possible outcomes.
Understanding discrete random variables helps the baker determine the likelihood of selling a certain number of batches, which in turn aids decision-making for daily operations.

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Most popular questions from this chapter

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. Construct the probability distribution function (PDF). $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline \\ \hline {} \\ \hline \\\ \hline \\ \hline {} \\ \hline \\ \hline {} \\ \hline \\ \hline \\\ \hline\end{array}$$

Use the following information to answer the next six exercises: On average, eight teens in the U.S. die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age. Is it likely that there will be more than 20 teens killed from motor vehicle injuries on any given day in the U.S.? Justify your answer numerically.

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Define the random variable \(X\).

A manufacturer of Christmas tree light bulbs knows that 3% of its bulbs are defective. Find the probability that a string of 100 lights contains at most four defective bulbs using both the binomial and Poisson distributions.

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. Approximately 8% of students at a local high school participate in after- school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-school sports all four years of high school. 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. How many seniors are expected to have participated in after-school sports all four years of high school? e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. f. Based upon numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically.
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