Question

Question: What is the expected value of the sum of the numbers appearing on two fair dice when they are rolled given that the sum of these numbers is at least nine. That is, what is \(E(X|A) \)where X is the sum of the numbers appearing on the two dice and A is the event that\(X \ge 9\)?

Short answer

Answer:

The expected value of the sum of the numbers appearing on two fair dice when they are rolled is 10.

Step-by-step solution

Note the given data

Given two dice and the sum is at least 9.

Definition  

The conditional expectation of the random variable X given the event A from the sample space S is \(E(X|A) = \sum\limits_{r \in X(s)} {r.} P(X = r|A)\)

Calculation

Given that the sum is at least 9. There are four possible outcomes (sum of 9, 10, 11, 12) with relative probabilities of 4,3,2,and 1

Therefore

P(X = 9) = 0.4

P(X = 10) = 0.3

P(X = 11) = 0.2

P(X = 12) = 0.1

Conditional expectation is

\(\begin{array}{l}E(X|A) = \sum\limits_{r \in X(s)} {r.} P(X = r|A)\\\begin{array}{*{20}{c}}{}&{}&{}\end{array} = 9 \times 0.4 + 10 \times 0.3 + 11 \times 0.2 + 12 \times 0.1\\\begin{array}{*{20}{c}}{}&{}&{}\end{array} = 10\end{array}\)

Conclusion  

The expected value of the sum of the numbers appearing on two fair dice when they are rolled is 10.