Q.3.28 - A First Course in Probability

Question

Prove or give a counterexample. If $E_1$ and $E_2$ are independent, then they are conditionally independent given $F$. 

Step-by-Step Solution

Answer

Rolling two independent six sided dice.

$E_1$ - the number on the first die is $1$,$E_2$ - the number on the second die is $4$, $F$ - the sum of the numbers is on both dice is $5$.

Step 1: Given Information

Independent events $E_1,E_2$.

Event $F$.

Step 2: Explanation

Do $E_1$ and $E_2$ have to be conditionally independent given $F$.

Not true

Counterexample: Rolling two independent six sided dice.$E_1$ - the number on the first die is $1$, $E_2$- the number on the second die is $4$, $F$ - the sum of the numbers is on both dice is 5 .

$P\left(E_1\right)=\frac{1}{6}$

$P\left(E_2\right)=\frac{1}{6}$

$P\left(E_1{E}_2\right)=\frac{1}{36}$

$P\left(E_2{E}_1\right)=P\left(E_1\right)P\left(E_2\right)$

Step 3: Explanation

$P\left(F\right)=\frac{4}{36}=\frac{1}{9}$

$P\left(E_1\mid F\right)=\frac{1}{4}$

$P\left(E_2\mid E_{1}F\right)=1$

$\left(E_2\mid E_{1}F\right)\ne P\left(E_1\mid F\right)$

Since the equality $P\left(E_2\mid E_{1}F\right)\ne P\left(E_1\mid F\right)$ is the definition of conditional independence these events are not conditionally independent.

Step 4: Final Answer

Rolling two independent six sided dice.

$E_1$- the number on the first die is $1$, $E_2$ - the number on the second die is $4$, $F$ - the sum of the numbers is on both dice is $5$.