Question

Question: What is the expected sum of the numbers that appear when three fair dice are rolled?

Short answer

Answer:

The expected sum is\(10.5\).

Step-by-step solution

Given information

Three fair dice are rolled.

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 \(10.5\).