Question

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?

A = the number of heads is odd, B = the number of tails is odd.

Answer 1

\[S = \left[ \left( H H H \right) \left( H H T \right) \left( H T H \right) \left( H T T \right) \left( T H H \right) \left( T H T \right) \left( T T H \right) \left( T T T \right) \right]\]

\[ P\left( A \right) = \frac{4}{8} = \frac{1}{2}\]
\[P\left( B \right) = \frac{4}{8} = \frac{1}{2}\]
\[\text{ Now } , \]
\[P\left( A \cap B \right) = \frac{0}{8} = 0\]
\[ P\left( A \cap B \right) \neq P\left( A \right)P\left( B \right)\]
\[\text{ Thus, A and B are not independent events. } \]