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Chapter 11: Problem 2

A die is rolled. The set of equally likely outcomes is \(\\{1,2,3,4,5,6\\}\). Find the probability of rolling a \(5 .\)

Short Answer

Expert verified
The probability of rolling a 5 is \(\frac{1}{6}\).

Step by step solution

01

Understand the total possible outcomes

When a fair die is rolled, there are 6 equally likely outcomes, represented by the set {1,2,3,4,5,6}. The total number of outcomes is 6.
02

Understand the event

The event we are considering is rolling a 5, which is a single outcome out of the total. There is only 1 outcome that satisfies this condition.
03

Calculate the probability

The probability of an event is calculated as the number of ways the event can occur divided by the total number of outcomes. So, the probability \(P\) of rolling a 5 is: \(P(5) = \frac{number \ of \ ways \ to \ roll \ a \ 5}{total \ number \ of \ outcomes} = \frac{1}{6}\).

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Key Concepts

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

Equally Likely Outcomes
When we talk about equally likely outcomes, we mean each outcome has the same chance of occurring. Imagine rolling a fair six-sided die. Each side, numbered from 1 to 6, is an outcome. Since the die is fair, each number from 1 to 6 has the same probability of showing up. This means each of the six outcomes is equally probable.

To see this in terms of probability, if there is no bias or trickery involved, each of these numbers appears exactly 1 out of 6 times, assuming many, many rolls. Whenever you're dealing with equally likely outcomes, calculating probabilities becomes straightforward, as we'll see in the coming sections.
Event Probability
Event probability is the likelihood of a specific event happening out of all possible outcomes. In probability, an event is something that we are interested in, like rolling a specific number on a die.

To find the probability of an event, you look at how many favorable outcomes match your event out of the total possible outcomes. If you're curious about the chances of rolling a 5 with a single die roll, this is your event probability.

In this case, because there is only one 5 on the die, there's just one favorable outcome. Understanding event probability helps you see how likely a desired result is in a given situation.
Dice Probability
Dice probability is a specific application of the general principles of probability. With a standard die, you have six faces, and each face represents a different outcome.

To calculate the probability of any single number, like a 5, appearing, you consider the number of sides:
  • There are 6 sides in total.
  • The number 5 appears on exactly 1 of these sides.
Thus, the probability of getting a 5 is determined by how often it shows up versus the total number of possibilities. In probability terms, you divide the number of ways to get your event (1 for the number 5) by the total number of outcomes (6).

This is a good example of how straightforward dice probability can be, thanks to the concept of equally likely outcomes.
Probability Calculation
When calculating probability, you use a simple formula:
  • Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes).
In our die example, the probability of rolling a 5 is calculated by dividing the number of favorable outcomes (1, since there's only one 5) by the total number of outcomes (6, as there are 6 numbers on the die).

Putting it into the formula, you have: \[ P(5) = \frac{1}{6} \]This tells you that for every roll, there's a one-sixth chance of rolling a 5. It's a systematic way to determine the likelihood of an event occurring in any situation with defined outcomes.

Understanding this basic calculation helps you with more complex probability problems, laying the foundation for statistical thinking.

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Most popular questions from this chapter

A social security number contains nine digits, such as 074-66-7795. How many different social security numbers can be formed?

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The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR MEDICAL INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \mathbf{\$ 2 0 , 0 0 0 )} \end{array} & \text { Probability } \\ \hline \$ 0 & 0.70 \\ \hline \$ 20,000 & 0.20 \\ \hline \$ 40,000 & 0.06 \\ \hline \$ 60,000 & 0.02 \\ \hline \$ 80,000 & 0.01 \\ \hline \$ 100,000 & 0.01 \\ \hline \end{array} $$
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