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.

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 \(10.5\).