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  ### 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 $$10.5$$.