Question
Let $$A, B, C$$ be three mutually independent events. Consider the two statements $${S_1}$$ and $${S_2}$$
$${S_1}$$ : $$A$$ and $$B$$ $$ \cup $$ $$C$$ are independent
$${S_2}$$ : $$A$$ and $$B$$ $$ \cap $$ $$C$$ are independent
Then,
A.
Both $${S_1}$$ and $${S_2}$$ are true
B.
Only $${S_1}$$ is true
C.
Only $${S_2}$$ is true
D.
Neither $${S_1}$$ nor $${S_2}$$ is true
Answer :
Both $${S_1}$$ and $${S_2}$$ are true
Solution :
$$\eqalign{
& P\left[ {A \cap \left( {B \cup C} \right)} \right] = P\left[ {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right] \cr
& = P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {A \cap B \cap C} \right) \cr
& = P\left( A \right)P\left( B \right) + P\left( A \right)P\left( C \right) - P\left( A \right)P\left( B \right)P\left( C \right) \cr
& = P\left( A \right)\left[ {P\left( B \right) + P\left( C \right) - P\left( {B \cap C} \right)} \right] = P\left( A \right)P\left( {B \cup C} \right) \cr
& \therefore \,\,{S_1}\,{\text{is true}}{\text{.}} \cr
& P\left( {A \cap \left( {B \cap C} \right)} \right) = P\left( A \right)P\left( B \right)P\left( C \right) = P\left( A \right)P\left( {B \cap C} \right) \cr
& \therefore \,\,{S_2}\,{\text{is also true}}{\text{.}} \cr} $$