# 4.3 Independence & Causality

## Mutually Independent Dice Rolls

Consider the following events the rolling 3 dice:

\begin{align}A_i&:=\text{the first die is a \(i, for $$i\in[1,6]$$}&\\B_i&:=\text{the second die is a $$i$$, for $$i\in[1,6]$$}&\\C_i&:=\text{the third die is a $$i$$, for $$i\in[1,6]$$}&\\S_i&:=\text{sum of the dice is $$i$$, for $$i\in[3,18]$$}&\end{align}\)

Which of the following events are mutually independent?

Exercise 1

Three dice throws are mutually independent. Any pair of dice throws is also independent. $$\Pr[A_1 \cap S_3]= \Pr[A_1 \cap B_1 \cap C_1]=\frac{1}{216}\neq \frac{1}{6}\cdot\frac{1}{216}=\Pr[A_1]\Pr[S_3].$$ $$\Pr[A_1 \cap B_1 \cap S_7]= \Pr[A_1 \cap B_1 \cap C_5]=\frac{1}{216}\neq \frac{1}{6}\cdot\frac{1}{6}\cdot\frac{15}{216}= \Pr[A_1]\Pr[B_1]\Pr[S_7].$$ $$\Pr[A_1 \cap B_1 \cap S_8]= \Pr[A_1 \cap B_1 \cap C_6]=\frac{1}{216}\neq \frac{1}{6}\cdot \frac{1}{6}\cdot\frac{21}{216}=\Pr[A_1]\Pr[B_1]\Pr[S_8].$$