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Chapter 4: Problem 9

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. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {0.15} \\ \hline 2 & {0.35} \\ \hline 3 & {0.40} \\ \hline 4 & {0.10} \\ \hline\end{array}$$ On average, how many batches should the baker make?

Short Answer

Expert verified
The baker should make about 2.45 batches on average.

Step by step solution

01

Understand the Problem

The problem provides us with a probability distribution for the number of batches of muffins (1, 2, 3, and 4). Our task is to find the expected value, which represents the average number of batches the baker should make to sell every muffin and no fewer.
02

Calculate Expected Value

The expected value of a random variable is calculated using the formula \( E(X) = \sum{x_i \cdot P(x_i)} \), where \( x_i \) are the possible values and \( P(x_i) \) are their respective probabilities.
03

Multiply Each Value by Its Probability

For each batch size, multiply the number of batches \( x_i \) by its probability \( P(x_i) \):- \( 1 \cdot 0.15 = 0.15 \)- \( 2 \cdot 0.35 = 0.70 \)- \( 3 \cdot 0.40 = 1.20 \)- \( 4 \cdot 0.10 = 0.40 \)
04

Sum the Products

Add up all the products from the previous step to find the expected value:\( E(X) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45 \)

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Key Concepts

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

Understanding Probability Distribution
Imagine a probability distribution as a way of showing how likely different outcomes are for a particular scenario. In this case, our baker is deciding how many batches of muffins to bake based on different observed probabilities. Each possible number of batches (1, 2, 3, or 4) has a specific likelihood of occurring, represented as a probability and adding up to 1.
  • The probability for 1 batch is 0.15.
  • The probability for 2 batches is 0.35.
  • The probability for 3 batches is 0.40.
  • The probability for 4 batches is 0.10.
This distribution helps the baker make informed decisions, guiding him towards the most probable amount.
Probability distributions are powerful tools in probability theory as they offer a visual and mathematical guide to future outcomes.
Exploring Random Variables
Random variables are a crucial part of understanding situations involving chance. They are essentially numerical representations of the outcomes from a random process, such as our baker's decisions on muffins. In this example, the random variable \(X\) represents the number of batches of muffins the baker can prepare.
  • Each possible number of batches (1, 2, 3, or 4) is a value that \(X\) can take.
  • Random variables can be discrete, like in our exercise where only a few specific numbers are possible.
These variables make it easier to model and predict real-world events, allowing businesses like our bakery to plan ahead efficiently by calculating the expected value.
Introduction to Probability Theory
Probability theory is the branch of mathematics that studies the unpredictability of random events. It's the foundation for figuring out how likely certain outcomes are, based on known information. In the context of the baker's problem, probability theory provides the framework to calculate the expected value, allowing the baker to determine the average number of batches he should bake.
  • It covers the principles of assigning probabilities to different events.
  • It uses mathematical formulas to express these probabilities, like the expected value formula used in the solution.
By leveraging probability theory, the baker can use these calculations to minimize waste and maximize sales, ensuring he meets customer demand without excess.

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Most popular questions from this chapter

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. Find the probability that Javier volunteers for at least one event each month. \(P(x>0)=\)________

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 chance of an IRS audit for a tax return with over \(\$ 25,000\) in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent. 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 audits are expected in a 20-year period? e. Find the probability that a person is not audited at all. f. Find the probability that a person is audited more than twice.

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? \(P(x=1)=\)_________

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. Six different colored dice are rolled. Of interest is the number of dice that show a one. 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 dice would you expect to show a one? e. Find the probability that all six dice show a one. f. Is it more likely that three or that four dice will show a one? Use numbers to justify your answer numerically.

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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