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Chapter 9: Problem 5

What is the expected sum of one roll of three fair six-sided dice?

Short Answer

Expert verified
The expected sum is 10.5.

Step by step solution

01

Calculate Expectation for One Die

The expected value for one fair six-sided die is the average of all possible outcomes. The possible outcomes are 1, 2, 3, 4, 5, and 6. So, we calculate the mean of these numbers: \( (1+2+3+4+5+6)/6 = 3.5 \).
02

Extend to Three Dice

The expected value of independent events such as the roll of three separate dice is the sum of their individual expected values. Since each die has an expected value of 3.5, the expected sum for three dice will be \( 3.5 + 3.5 + 3.5 = 10.5 \).

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Key Concepts

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

Probability
Probability is a way of measuring how likely an event is to occur. It's a fundamental concept in statistics and helps us understand random events like rolling dice. In its simplest form, probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. For instance, when you roll a six-sided die, there are six possible outcomes.
  • If you're interested in what probability a single face, such as getting a 4; its probability is 1/6, since there is one '4' and six possible faces.
  • If you want to know the probability of rolling an even number, you can add the probabilities of rolling a 2, 4, or 6, resulting in a probability of 3/6 or 1/2.
Probability helps us calculate various possibilities and can be used to determine expected outcomes, like the expected sum when rolling dice.
Six-Sided Dice
A six-sided die, also known as a d6, is a cube with each of its faces numbered from 1 to 6. It is commonly used in a variety of games and probability exercises. Each face of a six-sided die has an equal chance of landing face up.
  • This means that, when you roll a single six-sided die, the probability of landing on any one number, from 1 to 6, is equal 鈥� exactly 1/6.
  • Six-sided dice are used in many probability puzzles and problems due to their straightforward nature and equal distribution of outcomes.
Understanding how to calculate expected values from a six-sided die involves computing the average of its possible outcomes, which is central to solving dice-related probability problems.
Independent Events
Independent events are situations where the outcome of one event does not influence the outcome of another. This is an important concept when dealing with probability.
For example, when rolling dice, each die roll is independent of the others. The result of rolling one die will not affect the result of another die.
  • This concept is crucial in calculating probabilities and expected values for rolling multiple dice.
  • If you roll multiple dice, each roll must be considered independently, and their expected values can be added together.
When working with independent events like rolling multiple six-sided dice, it's important to remember that each die has its own probability distribution, and we can simply add the expected values of each die to find the total expected value.

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Most popular questions from this chapter

Verify that \(\int_{1}^{\infty} e^{-x / 2} d x=2 / \sqrt{e}\).

A thin plate lies in the region between the circle \(x^{2}+y^{2}=4\) and the circle \(x^{2}+y^{2}=1\), above the \(x\) -axis. Find the centroid.

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Verify that \(\pi \int_{0}^{1}(1+\sqrt{y})^{2}-(1-\sqrt{y})^{2} d y+\pi \int_{1}^{4}(1+\sqrt{y})^{2}-(y-1)^{2}=\frac{8}{3} \pi+\frac{65}{6} \pi=\frac{27}{2} \pi\).

Verify that \(\int_{0}^{1} \pi\left(1-x^{2}\right)^{2} d x=\frac{8}{15} \pi\).
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