/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Suppose that \(5 \%\) of cereal ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 85

Suppose that \(5 \%\) of cereal boxes contain a prize and the other \(95 \%\) contain the message, "Sorry, try again." Consider the random variable \(x,\) where \(x=\) number of boxes purchased until a prize is found. a. What is the probability that at most two boxes must be purchased? b. What is the probability that exactly four boxes must be purchased? c. What is the probability that more than four boxes must be purchased?

Short Answer

Expert verified
a. The probability of at most two boxes needing to be purchased is \(P(X \leq 2) \approx 0.0975\). b. The probability that exactly four boxes must be purchased is \(P(X = 4) 鈮� 0.043\). c. The probability that more than four boxes must be purchased is \(P(X > 4) 鈮� 0.8144\).

Step by step solution

01

- Recalling the Geometric Distribution Formula

The formula for the geometric probability distribution is given by: \[P(X = k) = (1-p)^{(k-1)}p\] Here, \(X\) is the random variable representing the number of trials until the first success, \(k\) is the specific number of trials we want to find the probability for, and \(p\) is the probability of success. In this problem, we have \(p=0.05\) and \(1-p=0.95\).
02

- Calculate the probability of at most two boxes

For part a, we want to find the probability that at most two boxes must be purchased. This means we want to find: \[P(X \leq 2) = P(X=1) + P(X=2)\] Using the geometric distribution formula from Step 1, we have: \[P(X=1) = (1-0.05)^{(1-1)}0.05 = 1^{0}0.05 = 0.05\] \[P(X=2) = (1-0.05)^{(2-1)}0.05 = 0.95^{1}0.05 \approx 0.0475\] Therefore, the probability of at most two boxes needing to be purchased is: \[P(X \leq 2) 鈮� 0.05 + 0.0475 = 0.0975\]
03

- Calculate the probability of exactly four boxes

For part b, we want to find the probability that exactly four boxes must be purchased. This means we want to find: \[P(X = 4)\] Using the geometric distribution formula from Step 1, we have: \[P(X=4) = (1-0.05)^{(4-1)}0.05 = 0.95^{3}0.05 \approx 0.043\] Therefore, the probability that exactly four boxes must be purchased is: \[P(X = 4) 鈮� 0.043\]
04

- Calculate the probability of more than four boxes

For part c, we want to find the probability that more than four boxes must be purchased. This means we want to find: \[P(X > 4) = 1 - P(X \leq 4)\] Because \(X=1\), \(X=2\), \(X=3\), and \(X=4\) are the only possibilities less than or equal to 4, we have: \[P(X \leq 4) = P(X=1) + P(X=2) + P(X=3) + P(X=4)\] We already found \(P(X=1)\), \(P(X=2)\), and \(P(X=4)\) in previous steps. To find \(P(X=3)\), we can use the geometric distribution formula: \[P(X=3) = (1-0.05)^{(3-1)}0.05 = 0.95^{2}0.05 \approx 0.0451\] So, the probability of purchasing four or fewer boxes is: \[P(X \leq 4) 鈮� 0.05 + 0.0475 + 0.0451 + 0.043 = 0.1856\] Finally, we can find the probability of purchasing more than four boxes: \[P(X > 4) = 1 - P(X \leq 4) 鈮� 1 - 0.1856 = 0.8144\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability of Success
Understanding the probability of success is crucial when working with geometric distributions. In layman's terms, it's like knowing the odds that you'll win a prize when you open a cereal box if only some of them have a surprise inside.

In the example with cereal boxes, we consider winning a prize as a 'success' and everything else a 'failure'. Here, since 5% of boxes have a prize, the probability of success, denoted as 'p', is 0.05 or 5%. This tiny number has a big role in our calculations. Every time you pick a box, you have a 5% chance of striking gold and finding that prize. If you keep getting 'Sorry, try again' messages, your chances don't suddenly get better 鈥� they stay at 5% for each new box.
Random Variable
In probability and statistics, a random variable is not just any number you think of; it's a way to represent the outcome of a 'random' process with numbers. It's like labeling each outcome of a game of chance with a score.

In our cereal box scenario, the random variable 'x' represents the number of cereal boxes you buy until you find a prize. So, if 'x' equals 1, that means you found a prize in the very first box. If 'x' is 3, you had to buy three boxes before getting lucky. This variable can take on any positive integer value, indicating the box number where your search for the prize ends. The key point here is that 'x' is not just any number; it鈥檚 the specific point in your sequence of tries where you finally win.
Probability Distribution
The term 'probability distribution' might sound intimidating, but it's simply a way to list all the possible outcomes of a random event and the likelihood of each. Think of it as a menu of all the potential results of your cereal box adventure, along with the chances of each happening.

In the context of geometric distribution, it features a peculiar characteristic 鈥� the memoryless property. This means regardless of how many 'Sorry' boxes you've already opened, your chances of opening a 'prize' box remain constant. It distributes the probability across an infinite number of possible outcomes, indicating that you could theoretically keep opening boxes forever until you find a prize. To calculate these probabilities, formulas come into play, as seen in the cereal box problem, to provide exact numbers for these chances, paving the way to a proper understanding of geometric distribution in practice.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Purchases made at small "corner stores" were studied by the authors of the paper "Changes in Quantity, Spending, and Nutritional Characteristics of Adult, Adolescent and Child Urban Corner Store Purchases After an Environmental Intervention" (Preventive Medicine [2015]: 81-85). Corner stores were defined as stores that are less than 200 square feet in size, have only one cash register, and primarily sell food. After observing a large number of corner store purchases in Philadelphia, the authors reported that the average number of grams of fat in a corner store purchase was \(21.1 .\) Suppose that the variable \(x=\) number of grams of fat in a corner store purchase has a distribution that is approximately normal with a mean of 21.1 grams and a standard deviation of 7 grams. a. What is the probability that a randomly selected corner store purchase has more than 30 grams of fat? b. What is the probability that a randomly selected corner store purchase has between 15 and 25 grams of fat? c. If two corner store purchases are randomly selected, what it the probability that both of these purchases will have more than 25 grams of fat?

Suppose that \(25 \%\) of the fire alarms in a large city are false alarms. Let \(x\) denote the number of false alarms in a random sample of 100 alarms. Approximate the following probabilities: a. \(P(20 \leq x \leq 30)\) b. \(P(20

The Wall Street Journal (February 15,1972 ) reported that General Electric was sued in Texas for sex discrimination over a minimum height requirement of 5 feet, 7 inches. The suit claimed that this restriction eliminated more than \(94 \%\) of adult females from consideration. Let \(x\) represent the height of a randomly selected adult woman. Suppose that \(x\) is approximately normally distributed with mean 66 inches (5 ft. 6 in.) and standard deviation 2 inches. a. Is the claim that \(94 \%\) of all women are shorter than \(5 \mathrm{ft}\). 7 in. correct? b. What proportion of adult women would be excluded from employment as a result of the height restriction?

A grocery store has an express line for customers purchasing at most five items. Consider the random variable \(x=\) the number of items purchased by a randomly selected customer using this line. Make two tables that represent two different possible probability distributions for \(x\) that have the same standard deviation but different means.

The light bulbs used to provide exterior lighting for a large office building have an average lifetime of 700 hours. If lifetime is approximately normally distributed with a standard deviation of 50 hours, how often should all the bulbs be replaced so that no more than \(20 \%\) of the bulbs will have already burned out?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.