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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 $$10.5$$. ### Want to see more solutions like these? 