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Chapter 10: Problem 7

Find the probability of the given event. Rolling a 2 with a single die

Short Answer

Expert verified
The probability is \( \frac{1}{6} \).

Step by step solution

01

Understanding the Problem

We want to find the probability of rolling a 2 with a single six-sided die. A standard six-sided die has six faces, each showing a different number from 1 to 6.
02

Determine the Total Number of Outcomes

When rolling a single die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). So, the total number of possible outcomes is 6.
03

Identify the Favorable Outcome

The event we are interested in is rolling a 2. Since there is only one face with the number 2 on a die, there is 1 favorable outcome.
04

Calculate the Probability

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. Hence, the probability of rolling a 2 is given by \( P(2) = \frac{1}{6} \).

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Key Concepts

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

Single Die Roll
In the world of probability, a single die roll is one of the simplest events to consider. When you roll a standard six-sided die, each face of the die represents one possible outcome. These outcomes are numbered from 1 through 6.

It's essential to understand that each number on the die has an equal chance of landing face up, assuming the die is fair and not tampered with. Availability of equal chances for each face is a crucial assumption in all probability calculations involving dice. When we say 'single die roll,' it means rolling one die one time and observing the result. This scenario forms the basis for many basic probability exercises.
Favorable Outcomes
The concept of favorable outcomes is central to calculating probability. When you roll a die, the favorable outcomes depend on the specific event you are considering. For instance, if you're interested in rolling a 2, the event occurs only under a particular condition: when the die lands on the number 2.

In this event:
  • There is 1 face with the number 2.
  • This single face represents the favorable outcome.
Hence, there is only 1 favorable outcome for rolling a 2 with a six-sided die. Favorable outcomes are the outcomes that satisfy the condition of the event you are interested in. This concept helps define the success scenarios in the context of probability.
Total Number of Outcomes
To effectively compute probabilities, understanding the total number of outcomes is essential. When rolling a six-sided die, each of its 6 faces can land face up, making 6 possible outcomes in total.

The formula for probability involves dividing the number of favorable outcomes by the total number of possible outcomes. Hence, in this die-rolling scenario:
  • We have 6 as the total number of outcomes since there are 6 faces.
  • The total number ensures every possibility is accounted for in probability calculations.
Thus, total outcomes set the basis for determining how likely an event is, making it a fundamental concept in probability theory.

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Most popular questions from this chapter

In Problems \(21-26\), write the given repeating decimal as a quotient of integers. $$ 0.222 \ldots $$

Determine whether the given sequence converges. $$ \left\\{\frac{3 n-2}{6 n+1}\right\\} $$

The sequence $$ \left\\{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right\\} $$ is known to converge (albeit very slowly) to a number \(\gamma\) called Euler's constant. Calculate at least the first 10 terms of the sequence. Conjecture the limit of the sequence.

Prove that \(\left(\begin{array}{c}n \\ r-1\end{array}\right)+\left(\begin{array}{l}n \\\ r\end{array}\right)=\left(\begin{array}{c}n+1 \\ r\end{array}\right), \quad 0

Write the given repeating decimal as a quotient of integers. $$ 0.616161 \ldots $$
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