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  A new era for learning is coming soon
  
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  Found in: Page 492
  
  ### Discrete Mathematics and its Applications
  
  Book edition 7th
  Author(s) Kenneth H. Rosen
  Pages 808 pages
  ISBN 9780073383095
  
  # Question: What is the expected sum of the numbers that appear when three fair dice are rolled?
  
  The expected sum is$$10.5$$.
  
  See the step by step solution
  
  ## Step 1: Given information
  
  Three fair dice are rolled.
  
  ## Step 2: Theorem for expected number
  
  The expected number of successes when$$n$$ mutually independent Bernoulli trials are performed, where$$p$$ is the probability of success on each trial, is $$np$$.
  
  ## Step 3: Calculate the expected sum
  
  Let S be the random variable denoting the sum of numbers when a fair die is rolled.
  
  Let s1, s2, s3 denote the corresponding sums for the three dice.
  
  We have $$S{\rm{ }} = {\rm{ }}s1 + s2 + s3$$
  
  $$\begin{array}{l}E(s) = E(s1 + s2 + s3)\\ = E(s1) + E(s2) + E(s3)\end{array}$$
  
  The expectation of the sum is the sum of the expectation values for three dice.
  
  But since they all are fair; they all have equal expectation values.
  
  Hence, $$E(s) = 3E(s1)$$
  
  The outcomes for a single fair dice are$$1, 2, 3,4, 5 and 6$$ with probability $$\frac{1}{6}$$.
  
  So, $$E(x) = \sum\limits_{i = 1}^6 {X.P(X = i)}$$
  
  $$\begin{array}{l}E(s1) = \frac{{1 + 2 + 3 + 4 + 5 + 6}}{6}\\ = 3.5\\E(s) = 3 \times 3.5 = 10.5\end{array}$$
  
  Thus, the expected sum is $$10.5$$.
explanation: The text extract contains a math question and its corresponding answer. The question asks for the expected sum of the numbers that appear when three fair dice are rolled, and the answer provided is 10.5, along with a detailed step-by-step solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: • :00Days
  • :00Hours
  • :00Mins
  • 00Seconds
  A new era for learning is coming soon
  
  ### Select your language
  
  Suggested languages for you:
  
  Americas
  
  Europe
  
  Q8E
  
  Expert-verified
  Found in: Page 492
  
  ### Discrete Mathematics and its Applications
  
  Book edition 7th
  Author(s) Kenneth H. Rosen
  Pages 808 pages
  ISBN 9780073383095
  
  # Question: What is the expected sum of the numbers that appear when three fair dice are rolled?
  
  The expected sum is$$10.5$$.
  
  See the step by step solution
  
  ## Step 1: Given information
  
  Three fair dice are rolled.
  
  ## Step 2: Theorem for expected number
  
  The expected number of successes when$$n$$ mutually independent Bernoulli trials are performed, where$$p$$ is the probability of success on each trial, is $$np$$.
  
  ## Step 3: Calculate the expected sum
  
  Let S be the random variable denoting the sum of numbers when a fair die is rolled.
  
  Let s1, s2, s3 denote the corresponding sums for the three dice.
  
  We have $$S{\rm{ }} = {\rm{ }}s1 + s2 + s3$$
  
  $$\begin{array}{l}E(s) = E(s1 + s2 + s3)\\ = E(s1) + E(s2) + E(s3)\end{array}$$
  
  The expectation of the sum is the sum of the expectation values for three dice.
  
  But since they all are fair; they all have equal expectation values.
  
  Hence, $$E(s) = 3E(s1)$$
  
  The outcomes for a single fair dice are$$1, 2, 3,4, 5 and 6$$ with probability $$\frac{1}{6}$$.
  
  So, $$E(x) = \sum\limits_{i = 1}^6 {X.P(X = i)}$$
  
  $$\begin{array}{l}E(s1) = \frac{{1 + 2 + 3 + 4 + 5 + 6}}{6}\\ = 3.5\\E(s) = 3 \times 3.5 = 10.5\end{array}$$
  
  Thus, the expected sum is $$10.5$$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question and its corresponding answer. The question asks for the expected sum of the numbers that appear when three fair dice are rolled, and the answer provided is 10.5, along with a detailed step-by-step solution.
  
  Label: question-and-answer