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Chapter 11: Problem 20

What does the expected value for the outcome of the roll of a fair die represent?

Short Answer

Expert verified
The expected value of the outcome of a roll of a fair die is 3.5. This means that if we continue to roll a fair die, we should expect that the long-term average value of the outcomes will converge to 3.5.

Step by step solution

01

Understand the Die

A fair die has six faces each numbered from 1 to 6. Each face has an equal probability of being rolled, which is \(\frac{1}{6}\).
02

Calculate Each Outcome's Product with Its Probability

The expected value is calculated by summing all possible outcomes of the random variable, each multiplied by their probability. For a single roll of a fair die, this means calculating \(1*\frac{1}{6} + 2*\frac{1}{6} + 3*\frac{1}{6} + 4*\frac{1}{6} + 5*\frac{1}{6} + 6*\frac{1}{6}\).
03

Sum the Outcomes

Adding these 6 outcomes together gives the final expected value. So, the expected value is \(\frac{1}{6} + \frac{2}{3} + \frac{1}{2} + \frac{2}{3} + \frac{5}{6} + 1 = 3.5\)

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Key Concepts

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

Understanding Probability
Probability is a way of expressing the likelihood of an event occurring. It is a concept widely used in mathematics to predict the chances of different outcomes. For a particular event, probability values range from 0 to 1, where 0 means the event will not happen and 1 indicates certainty that it will happen.
To calculate the probability of an event, you divide the number of successful outcomes by the total number of possible outcomes. For instance, when rolling a fair die with six faces:
  • Each face has an equal chance of occurring.
  • The probability of rolling any specific number, say a 4, is \( \frac{1}{6} \) since there are six possible outcomes.
Understanding probability helps determine how likely or unlikely an event can be while considering all possible outcomes.
Exploring Random Variables
A random variable is a numerical value resulting from a random event or experiment. It assigns numerical values to each possible outcome of a random process. In the context of rolling a die, each face number from 1 to 6 is a potential outcome and, subsequently, a random variable. This makes the roll of a die a simple random experiment.
  • A random variable can be discrete or continuous. Rolling a die results in a discrete random variable because there are a finite number of possible outcomes.
  • Each random variable has an associated probability, which helps calculate expectations, such as the expected value for a die roll.
The concept of random variables is crucial in statistical analysis because it translates the qualitative outcomes of random processes into quantitative data for calculation and prediction.
Rolling a Fair Die
When we talk about a fair die, we refer to a six-sided die where each side has an equal chance of showing up when rolled. This is essential for assessing outcomes in probability and statistics since fairness ensures that each face generates the same probability during a roll. Understanding a fair die includes:
  • Each face contributing equally to the probability distribution, meaning \( P(X = x) = \frac{1}{6} \) for any outcome \( x \).
  • This uniform probability distribution is vital for calculating the expected value correctly. With equal probabilities, the expected value relies only on the average of all possible outcomes.
Rolling a fair die illustrates an ideal example of uniform probability and serves as a basis for learning more intricate statistical concepts.

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Most popular questions from this chapter

The probability that a region prone to flooding will flood in any single year is \(\frac{1}{10}\). a. What is the probability of a flood two years in a row? b. What is the probability of flooding in three consecutive years? c. What is the probability of no flooding for ten consecutive years? d. What is the probability of flooding at least once in the next ten years?

An ice chest contains six cans of apple juice, eight cans of grape juice, four cans of orange juice, and two cans of mango juice. Suppose that you reach into the container and randomly select three cans in succession. Find the probability of selecting a can of grape juice, then a can of orange juice, then a can of mango juice.

One card is randomly selected from a deck of cards. Find the odds against drawing a \(9 .\)

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I found the probability of getting rain at least once in ten days by calculating the probability that none of the days have rain and subtracting this probability from \(1 .\)
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