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Two fair 6-sided dice are rolled. What is the expected value of the sum of the dice? What is the
expected value of the product of the dice?

Sep 5, 2018

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Two fair 6-sided dice are rolled.

1. What is the expected value of the sum of the dice?

2. What is the expected value of the product of the dice?

1. What is the expected value of the sum of the dice?

$$\begin{array}{|r|r|r|r|r|r|r|} \hline & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 5 & 6 & 7 & 8 & 9 & 10& 11 \\ \hline 6 & 7 & 8 & 9 & 10& 11& 12 \\ \hline \end{array}$$

$$\begin{array}{|lrcll|} \hline \text{Expected value}\\ = \dfrac{1\cdot 2 + 2\cdot 3+ 3\cdot 4+ 4\cdot 5+ 5\cdot 6 + 6\cdot 7+ 5\cdot 8+ 4\cdot 9+ 3\cdot 10+ 2\cdot 11 + 1\cdot12}{36} \\ = \dfrac{252}{36} \\\\ = 7 \\ \hline \end{array}$$

2. What is the expected value of the product of the dice?

$$\begin{array}{|r|r|r|r|r|r|r|} \hline & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 2 & 2 & 4 & 6 & 8 & 10& 12 \\ \hline 3 & 3 & 6 & 9 & 12& 15& 18 \\ \hline 4 & 4 & 8 & 12& 16& 20& 24 \\ \hline 5 & 5 & 10& 15& 20& 25& 30 \\ \hline 6 & 6 & 12& 18& 24& 30& 36 \\ \hline \end{array}$$

$$\begin{array}{|lrcll|} \hline \text{Expected value}\\ =\dfrac{ 6\cdot \dfrac{1+6}{2}+6\cdot \dfrac{2+12}{2} + 6\cdot \dfrac{3+18}{2}+ 6\cdot \dfrac{4+24}{2}+ 6\cdot \dfrac{5+30}{2}+ 6\cdot \dfrac{6+36}{2} } {36}\\ = \dfrac{441}{36} \\\\ = 12.25 \\ \hline \end{array}$$

Sep 6, 2018