Problem 11
Question
Write each event in set notation. Give the probability of the event. A student gives the answer to a probability problem as \(\frac{6}{5}\). Explain why this answer must be incorrect.
Step-by-Step Solution
Verified Answer
The probability \( \frac{6}{5} \) is incorrect because probabilities must be between 0 and 1.
1Step 1: Understanding Set Notation
In probability, an event can be described as a subset of a sample space. For example, if we consider the event of rolling a die, the sample space is \( \{1, 2, 3, 4, 5, 6\} \). An example event could be rolling an even number, which can be written in set notation as \( \{2, 4, 6\} \). The probability of this event is determined by how many outcomes satisfy it.
2Step 2: Analyzing the Given Probability
The student claims the probability of an event is \( \frac{6}{5} \). Probability values are always between 0 and 1, inclusive. This means \(0 \leq P(A) \leq 1\) for any event \(A\). This is because probability is defined as the ratio of favorable outcomes to total possible outcomes.
3Step 3: Identifying the Error
Since \( \frac{6}{5} = 1.2 \), this value is greater than 1. A probability greater than 1 is not possible because it would imply more outcomes than exist in the sample space. Therefore, the student's answer of \( \frac{6}{5} \) is incorrect.
Key Concepts
Set NotationSample SpaceProbability ValuesFavorable Outcomes
Set Notation
Set notation is a fundamental concept in probability that helps clearly define events within a sample space. When we talk about an event in the context of probability, we refer to specific outcomes of an experiment. Let's use the example of rolling a six-sided die. The sample space, which represents all possible outcomes, is \(\{1, 2, 3, 4, 5, 6\}\).
An event is a subset of this sample space. For instance, if the event is rolling an even number, we can express this in set notation as \(\{2, 4, 6\}\). Each element within the curly brackets is an outcome that belongs to the event.
Using set notation makes it easier to visually comprehend and communicate these outcomes. Additionally, it helps in calculating probabilities by directly linking outcomes to their respective probabilities in the sample space.
An event is a subset of this sample space. For instance, if the event is rolling an even number, we can express this in set notation as \(\{2, 4, 6\}\). Each element within the curly brackets is an outcome that belongs to the event.
Using set notation makes it easier to visually comprehend and communicate these outcomes. Additionally, it helps in calculating probabilities by directly linking outcomes to their respective probabilities in the sample space.
Sample Space
The sample space is a critical element in probability, encompassing all possible outcomes of a given experiment. Picture it as a complete blueprint of everything that can occur when conducting a trial. For example, if you flip a coin, the sample space is \(\{\text{Heads}, \text{Tails}\}\).
In terms of the die-rolling scenario, the sample space is \(\{1, 2, 3, 4, 5, 6\}\), including each number on the die.
Understanding the sample space is vital because it serves as the foundation for determining probabilities. The probability of any event is calculated against the total number of outcomes in this sample space. Hence, defining the sample space accurately is the first step toward correct probability determination.
In terms of the die-rolling scenario, the sample space is \(\{1, 2, 3, 4, 5, 6\}\), including each number on the die.
Understanding the sample space is vital because it serves as the foundation for determining probabilities. The probability of any event is calculated against the total number of outcomes in this sample space. Hence, defining the sample space accurately is the first step toward correct probability determination.
Probability Values
Probability values quantify the likelihood of an event happening. They are always expressed as a number between 0 and 1, inclusive. A probability of 0 means the event cannot happen, while a probability of 1 means the event is certain to occur.
This is expressed by the inequality \(0 \leq P(A) \leq 1\), where \(P(A)\) denotes the probability of event \(A\).
If we say \(P(\text{rolling a 3}) = \frac{1}{6}\), we are indicating a one in six chance of rolling a 3 on a fair die. Whenever a probability given, like \(\frac{6}{5}\) or 1.2, exceeds 1, it signals an error. A proper understanding of how these values operate within the range ensures accurate calculations and interpretations.
This is expressed by the inequality \(0 \leq P(A) \leq 1\), where \(P(A)\) denotes the probability of event \(A\).
If we say \(P(\text{rolling a 3}) = \frac{1}{6}\), we are indicating a one in six chance of rolling a 3 on a fair die. Whenever a probability given, like \(\frac{6}{5}\) or 1.2, exceeds 1, it signals an error. A proper understanding of how these values operate within the range ensures accurate calculations and interpretations.
Favorable Outcomes
Favorable outcomes are specific results in the context of probability that satisfy the condition of the event we're interested in. They are essential in determining the probability of an event.
To compute this, we take the number of favorable outcomes and divide it by the total outcomes in the sample space. So, using the die example, if you want to compute the probability of rolling an even number (event \(\{2, 4, 6\}\)), there are three favorable outcomes (2, 4, and 6) out of six possible outcomes. Thus, \(P(\text{even}) = \frac{3}{6} = \frac{1}{2}\).
By understanding favorable outcomes, you ensure you calculate probabilities accurately, aligning with the theoretical foundations of probability theory.
To compute this, we take the number of favorable outcomes and divide it by the total outcomes in the sample space. So, using the die example, if you want to compute the probability of rolling an even number (event \(\{2, 4, 6\}\)), there are three favorable outcomes (2, 4, and 6) out of six possible outcomes. Thus, \(P(\text{even}) = \frac{3}{6} = \frac{1}{2}\).
By understanding favorable outcomes, you ensure you calculate probabilities accurately, aligning with the theoretical foundations of probability theory.
Other exercises in this chapter
Problem 10
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n-1}(n+1)$$
View solution Problem 11
Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$
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Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}8 \\\3\end{array}\right)$$
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CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$-4,-12,-36,-108, \dots$$
View solution