Q.3.27 - A First Course in Probability

Question

Extend the definition of conditional independence to more than $2$ events.

Step-by-Step Solution

Answer

Conditional independence is independence in conditional probability.

Step 1: Given Information

The definition of conditional independence to more than $2$ events. 

Step 2: Explanation

Conditional independence of two events $-E_1,E_2$, with condition $F$ is defined by any of two equivalent conditions:

$P\left(E_1\mid E_{2}F\right)=P\left(E_1\mid F\right) \iff P\left(E_1{E}_2\mid F\right)=P\left(E_1\mid F\right)P\left(E_2\mid F\right)$

Conditional independence is independence in conditional probability.

Generalize the formula for independence of multiple events:

$n$ events $E_1,E_2,\dots ,E_n$ are conditionally independent given $F$ if

for any $k$ and any different $j_1,j_2,\dots ,j_k\in \{1,2,\dots ,n\}$

$P\left({\cap }_{i=1}^k{E}_{j_i}\mid F\right)=\prod _{i=1}^{k}P\left(E_{j_i}\mid F\right)$

Step 3: Final Answer

Multiple events are conditionally independent if conditional probability of intersection of any subset of events is the product of the conditional probability of those events.