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Chapter 7: Problem 24

There are eight grades in Garfield Elementary School. If a student is selected at random from the school, then the probability that the student is in the first grade is \(\frac{1}{8}\)

Short Answer

Expert verified
The probability of a student being in the first grade at Garfield Elementary School, out of the eight possible grades, is calculated using the probability formula: P(First Grade) = \(\frac{1}{8}\), which means that there's a \(\frac{1}{8}\) chance of a randomly selected student being in the first grade.

Step by step solution

01

Understand the probability formula

The probability of an event happening is the number of favorable outcomes divided by the number of possible outcomes. In this case, the favorable outcome is the student being in the first grade, and the possible outcomes are the students being in any of the eight grades.
02

Identify the favorable and possible outcomes

In this problem, we have the following: - Favorable outcomes: Student is in the first grade (1 possible way) - Possible outcomes: Students can be in any of the eight grades (8 possible ways)
03

Use the probability formula to calculate

Now, we can use the probability formula. The probability of a student being in the first grade (P(First Grade)) is equal to the number of favorable outcomes divided by the number of possible outcomes: P(First Grade) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\)
04

Substitute the values

We can now substitute the values we identified in Step 2: P(First Grade) = \(\frac{1}{8}\)
05

Simplify and interpret the result

The probability of a student being in the first grade is \(\frac{1}{8}\). This means that if a student is selected at random from the school, there's a \(\frac{1}{8}\) chance that this student is in the first grade.

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Key Concepts

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

Probability Formula
The probability formula is at the heart of calculating chances in various situations, helping us determine how likely an event is to occur. To simplify, probability is the measure of the likelihood that an event will happen. Mathematically, it's given by the formula:

\[\begin{equation} Probability = \frac{Number\ of\ favorable\ outcomes}{Number\ of\ possible\ outcomes} \end{equation}\]
This formula is essential because it provides a clear and precise way to quantify uncertainty. In the context of our elementary school example, where we want to find the likelihood of a student being in the first grade, it very much simplifies to counting how many outcomes we favor (a student being in the first grade) versus all the things that could possibly happen (students being in any of the eight grades).
Favorable Outcomes
Favorable outcomes are those which satisfy the conditions of the event we're focused on. They are the 'wins' or 'successes' that we count when we're figuring out probability. For an outcome to be considered favorable, it must align exactly with the event we're interested in.

For example, in our school scenario, the favorable outcome is the specific condition we're checking for: A student being in the first grade. There's only one way this can happen--a student is simply in the first grade or not. Thus, we count one favorable outcome. It's essential to identify the favorable outcome clearly, as it defines the numerator in our probability formula and ultimately influences our understanding of how likely an event is.
Possible Outcomes
The concept of possible outcomes is akin to looking at every different result that could happen in a given situation. Unlike favorable outcomes, possible outcomes consider all conceivable scenarios, regardless of whether they match our condition of interest. They form the basis of the denominator in our probability formula.

In the illustration of Garfield Elementary School, the possible outcomes are any of the grades a student could be in, which are eight in total. When calculating probability, it is critical to account for all these potential situations to ensure the accuracy of our probability calculation. Remember, the probability is a fraction鈥� getting the denominator right is as important as the numerator, if not more so.

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Most popular questions from this chapter

An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome-heads or tails. The first coin is a two-headed coin, the second is a biased coin such that \(P(\mathrm{H})=.75\), and the third is a fair coin. a. What is the probability that the coin that is tossed will show heads? b. If the coin selected shows heads, what is the probability that this coin is the fair coin?

The Office of Admissions and Records of a large western university released the accompanying information concerning the contemplated majors of its freshman class:3 $$\begin{array}{lccc} \text { Major } & \text {This Major, \% } & \text { Females, \% } & \text { Males, \% } \\ \hline \text { Business } & 24 & 38 & 62 \\ \hline \text { Humanities } & 8 & 60 & 40 \\ \hline \text { Education } & 8 & 66 & 34 \\ \hline \text { Social science } & 7 & 58 & 42 \\ \hline \text { Natural sciences } & 9 & 52 & 48 \\ \hline \text { Other } & 44 & 48 & 52 \\ \hline \end{array}$$ What is the probability that a. A student selected at random from the freshman class is a female? b. A business student selected at random from the fresh- man class is a male? c. A female student selected at random from the freshman class is majoring in business?

Quaurr CoNrroL. An automobile manufacturer obtains the microprocessors used to regulate fuel consumption in its automobiles from three microelectronic firms: \(\mathrm{A}, \mathrm{B}\), and C. The quality-control department of the company has determined that \(1 \%\) of the microprocessors produced by firm \(A\) are defective, \(2 \%\) of those produced by firm \(B\) are defective, and \(1.5 \%\) of those produced by firm \(\mathrm{C}\) are defective. Firms \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) supply \(45 \%, 25 \%\), and \(30 \%\), respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?

According to a study of 100 drivers in metropolitan Washington D.C. whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) $$ \begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside } \mathrm{car} \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned} $$ If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?

A time study was conducted by the production manager of Universal Instruments to determine how much time it took an assembly worker to complete a certain task during the assembly of its Galaxy home computers. Results of the study indicated that \(20 \%\) of the workers were able to complete the task in less than \(3 \mathrm{~min}\), \(60 \%\) of the workers were able to complete the task in \(4 \mathrm{~min}\) or less, and \(10 \%\) of the workers required more than \(5 \mathrm{~min}\) to complete the task. If an assembly-line worker is selected at random from this group, what is the probability that a. He or she will be able to complete the task in 5 min or less? b. He or she will not be able to complete the task within 4 min? c. The time taken for the worker to complete the task will be between 3 and 4 min (inclusive)?
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