Question
A coin is tossed three times. Consider the following events :
$$A:$$ No head appears
$$B:$$ Exactly one head appears
$$C:$$ At least two heads appear
Which one of the following is correct ?
A.
$$\left( {A \cup B} \right) \cap \left( {A \cup C} \right) = B \cup C$$
B.
$$\left( {A \cap B'} \right) \cup \left( {A \cap C'} \right) = B' \cup C'$$
C.
$$A \cap \left( {B' \cup C'} \right) = A \cup B \cup C$$
D.
$$A \cap \left( {B' \cup C'} \right) = B' \cap C'$$
Answer :
$$A \cap \left( {B' \cup C'} \right) = B' \cap C'$$
Solution :
$$\eqalign{
& U = \left\{ {\left( {HHH} \right)\left( {HHT} \right)\left( {HTH} \right)\left( {HTT} \right)\left( {THH} \right)\left( {THT} \right)\left( {TTH} \right)\left( {TTT} \right)} \right\} \cr
& A = \left\{ {\left( {TTT} \right)} \right\} \cr
& B = \left\{ {\left( {HTT} \right)\left( {THT} \right)\left( {TTH} \right)} \right\} \cr
& C = \left\{ {\left( {HHH} \right)\left( {HHT} \right)\left( {HTH} \right)\left( {THH} \right)} \right\} \cr
& {\text{By checking the options,}}\left( {\bf{D}} \right){\text{is correct}} \cr
& A \cap \left( {B' \cup C'} \right) = B' \cap C' \cr} $$