# 4.3 Independence & Causality

## Independent Dice Rolls

Consider the following events for rolling 2 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 $$i\in[1,6]$$}&\\S_i&:=\text{sum of the dice is $$i$$, for $$i\in[2,12]$$}&\end{align}\)

Which of the following events are independent?

Exercise 1

Two dice throws are independent.

$$\Pr[A_3 \cap S_2]=\Pr[\emptyset]=0\neq \frac{1}{6}\cdot\frac{1}{36}=\Pr[A_3]\Pr[S_2].$$ $$\Pr[A_2 \cap S_6]= \Pr[A_2 \cap B_4]=\frac{1}{36}\neq \frac{1}{6}\cdot\frac{5}{36}=\Pr[A_2]\Pr[S_6].$$ $$\Pr[A_1 \cap S_7]= \Pr[A_1 \cap B_6]=\frac{1}{36} = \frac{1}{6}\cdot\frac{6}{36}=\Pr[A_1]\Pr[S_7].$$