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Chapter 5: Problem 58

In a sample of 275 students, 20 say they are vegetarians. Of the vegetarians, 9 eat both fish and eggs, 3 eat eggs but not fish, and 7 eat neither. Choose one of the vegetarians at random. What is the probability that the chosen student eats fish or eggs? (a) \(9 / 20\) (c) \(22 / 20\) (e) \(22 / 275\) (b) \(13 / 20\) (d) \(9 / 275\)

Short Answer

Expert verified
(b) \( \frac{13}{20} \)

Step by step solution

01

Identify Vegetarians

First, determine the total number of vegetarians from the sample. We are given that there are 20 vegetarians in total.
02

Categorize Eating Preferences

We categorize the vegetarians based on their eating preferences: 9 eat both fish and eggs, 3 eat eggs but not fish, and 7 eat neither fish nor eggs. Let the number of vegetarians who eat only fish be represented by x. Since this problem involves trying to find the probability of a vegetarian eating either fish or eggs, let's find all who eat either (or both).
03

Find Those Eating Fish or Eggs

To find the number of vegetarians who eat fish or eggs, we need to consider those who eat only fish, only eggs, and both fish and eggs. We already know there are 9 who eat both fish and eggs, 3 who eat only eggs, and we need to solve for x (those who only eat fish).The equation is: Total vegetarians = Those who eat fish only + Those who eat eggs only + Those who eat both + Those who eat neither:\[ 9 + 3 + 7 + x = 20 \]Solving for x, we find that:\[ x = 20 - 9 - 3 - 7 = 1 \]So, 1 vegetarian eats only fish.
04

Calculate the Total for Fish or Eggs

Add the number of people who eat both, only fish, or only eggs to determine the total who eat fish or eggs:\[ 9 \text{ (both)} + 3 \text{ (only eggs)} + 1 \text{ (only fish)} = 13 \]
05

Calculate the Probability

Given that there are 13 vegetarians who either eat fish or eggs, divide this number by the total number of vegetarians to get the probability.\[ P = \frac{13}{20} \]
06

Select the Correct Answer

Comparing our answer with the options given, we see that option (b) matches our calculated probability. Thus, the correct answer is option (b) \( \frac{13}{20} \).

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Key Concepts

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

Sample Space
The concept of sample space is fundamental in probability theory. It refers to the set of all possible outcomes of a particular experiment or activity. In our exercise, the sample space is relatively straightforward. We are dealing with a group of 20 vegetarians. Each vegetarian represents a unique outcome, meaning the sample space includes all 20 vegetarians.
When solving a probability problem, always start by defining the sample space clearly. This allows you to consider all potential outcomes.
  • For example, if an experiment involved rolling a six-sided die, the sample space would consist of the numbers 1 through 6.
  • In our case, where we are selecting a vegetarian at random, the sample space is the list of all 20 vegetarians.
Event
An event in probability refers to a specific set of outcomes that we are interested in. In our exercise, the event of interest is a vegetarian eating either fish or eggs. This event is composed of outcomes where vegetarians eat either or both of these items. It's important to correctly identify the event as it determines what subset of the sample space you are working with.
For this exercise:
  • We identified three groups contributing to the event: those eating only fish, only eggs, and both fish and eggs.
  • This meant looking at specific preferences: 1 eats only fish, 3 eat only eggs, and 9 eat both fish and eggs.

Determining the probability of an event involves calculating the number of favorable outcomes over the total number of possible outcomes (from the sample space).
Conditional Probability
Conditional probability deals with finding the probability of an event occurring, given that another event has already occurred. In many problems, understanding this can bring clarity.
In the exercise, although we don't explicitly use conditional probability, the overall approach benefits from understanding related influences. For instance, let's imagine another question asking about the probability of a vegetarian eating eggs given they eat fish, or vice-versa.
The formula for conditional probability is:
  • \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
This expression means 'the probability of A given B.' Remember:
  • This can help refine probabilities within subsections of a population.
  • Ensures accuracy in more complex exercises, where conditions influence outcomes.
Complement Rule
The complement rule in probability is a valuable tool for solving a wide range of problems. It states that the probability of an event not occurring is one minus the probability of the event occurring.
If we're considering an event A, then its complement is all outcomes that are not A.
  • \[ P(A^c) = 1 - P(A) \]
In the context of our exercise, if we wanted to find the probability of a vegetarian not eating either fish or eggs, we could apply the complement rule.
  • First, determine the probability of vegetarians eating fish or eggs (which we've calculated as \( \frac{13}{20} \)).
  • Then, subtract this probability from 1 to find the probability of the complement event: no fish or eggs.

Using the complement rule efficiently can often simplify calculations, particularly when the non-occurrence of an event is easier to calculate.

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Most popular questions from this chapter

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, \(50 \%\) of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets \(95 \%\) of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table \(D .\) If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Simulation blunders Explain what's wrong with each of the following simulation designs. (a) According to the Centers for Disease Control and Prevention, about \(36 \%\) of U.S. adults were obese in \(2012 .\) To simulate choosing 8 adults at random and seeing how many are obese, we could use two digits. Let 00 to 35 represent obese and 36 to 99 represent not obese. Move across a row in Table \(D,\) two digits at a time, until you find 8 distinct numbers (no repeats). Record the number of obese people selected. (b) Assume that the probability of a newborn being a boy is \(0.5 .\) To simulate choosing a random sample of 9 babies who were born at a local hospital today and observing their gender, use one digit. Use rand Int (0,9) on your calculator to determine how many babies in the sample are male.

Taking the train According to New Jersey Transit, the 8: 00 A.M. weekday train from Princeton to New York City has a \(90 \%\) chance of arriving on time. To test this claim, an auditor chooses 6 weekdays at random during a month to ride this train. The train arrives late on 2 of those days. Does the auditor have convincing evidence that the company's claim isn't true? Design and carry out a simulation to estimate the probability that a train with a \(90 \%\) chance of arriving on time each day would be late on 2 or more of 6 days. Follow the four-step process.

Simulation blunders Explain what's wrong with each of the following simulation designs. (a) A roulette wheel has 38 colored slots -18 red, 18 black, and 2 green. To simulate one spin of the wheel, let numbers 00 to 18 represent red, 19 to 37 represent black, and 38 to 40 represent green. (b) About \(10 \%\) of U.S. adults are left-handed. To simulate randomly selecting one adult at a time until you find a left-hander, use two digits. Let 00 to 09 represent being left-handed and 10 to 99 represent being right- handed. Move across a row in Table \(D\), two digits at a time, skipping any numbers that have already appeared, until you find a number between 00 and \(09 .\) Record the number of people selected.

A basketball player claims to make \(47 \%\) of her shots from the field. We want to simulate the player taking sets of 10 shots, assuming that her claim is true. Twenty-five repetitions of the simulation were performed. The simulated number of makes in each set of 10 shots was recorded on the dotplot below. What is the approximate probability that a \(47 \%\) shooter makes 5 or more shots in 10 attempts? (a) \(5 / 10\) (b) \(3 / 10\) (c) \(12 / 25\) (d) \(3 / 25\) (e) \(47 / 100\)
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